Pacific Journal of Mathematics Vol 234 Issue 1, Jan 2008

Volume 234

No. 1

January 2008

Pacific Journal of Mathematics

PACIFIC JOURNAL OF MATHEMATICS


2008 Vol. 234, No. 1 Volume 234

No. 1

January 2008

PACIFIC JOURNAL OF MATHEMATICS http://www.pjmath.org Founded in 1951 by E. F. Beckenbach (1906–1982)

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Sorin Popa Department of Mathematics University of California Los Angeles, CA 90095-1555 [email protected]

Robert Finn Department of Mathematics Stanford University Stanford, CA 94305-2125 [email protected]

Jiang-Hua Lu Department of Mathematics The University of Hong Kong Pokfulam Rd., Hong Kong [email protected]

Jie Qing Department of Mathematics University of California Santa Cruz, CA 95064 [email protected]

Kefeng Liu Department of Mathematics University of California Los Angeles, CA 90095-1555 [email protected]

Alexander Merkurjev Department of Mathematics University of California Los Angeles, CA 90095-1555 [email protected]

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PACIFIC JOURNAL OF MATHEMATICS Vol. 234, No. 1, 2008

CATEGORY O FOR THE VIRASORO ALGEBRA: COHOMOLOGY AND KOSZULITY B RIAN D. B OE , DANIEL K. NAKANO

AND

E MILIE W IESNER

We investigate blocks of the Category O for the Virasoro algebra over C. We demonstrate that the blocks have Kazhdan–Lusztig theories and that the truncated blocks give rise to interesting Koszul algebras. The simple modules have BGG resolutions, and from this we compute the extensions between Verma modules and simple modules, and between pairs of simple modules.

1. Introduction Bernˇste˘ın, Gel0 fand, and Gel0 fand [1976] initiated the study of Category O for complex semisimple Lie algebras. Since the introduction of Category O, much progress has been made in studying the structure of blocks for this category and its variants. One of the major results in this area was the formulation and proof of the Kazhdan–Lusztig (KL) conjectures [Kazhdan and Lusztig 1979; Be˘ılinson and Bernstein 1981; Brylinski and Kashiwara 1981], which provide a recursive formula for the characters of simple modules in Category O. These conjectures have been equivalently formulated in terms of Ext-vanishing conditions between simple modules and Verma modules. For semisimple algebraic groups over fields of positive characteristic p > 0, an analogous conjecture has been provided by Lusztig as long as p is at least as large as the Coxeter number of the underlying root system. The characteristic p Lusztig Conjecture still remains open. In an attempt to better understand both the original Kazhdan–Lusztig Conjecture and the Lusztig Conjecture, Cline, Parshall, and Scott [1988; 1993; 1997] developed an axiomatic treatment of highest weight categories with the added structures involving “Kazhdan–Lustzig theories” and Koszulity. Irving [1990; 1992] has partially developed some theories along these lines. Be˘ılinson, Ginzburg, and Soergel [1996] proved that the principal block of Category O is Koszul using perverse sheaves and established Koszul duality between various blocks of Category O, MSC2000: 17B68, 17B10, 17B56, 17B65. Keywords: Virasoro algebra, Lie algebra cohomology, extensions. Boe’s research was partially supported by NSA grant H98230-04-1-0103. Nakano’s research was partially supported by NSF grant DMS-0400548. 1

2

BRIAN D. BOE, DANIEL K. NAKANO

AND

EMILIE WIESNER

which provides an alternative proof of the KL Conjecture. The work of Cline, Parshall, and Scott is important because it isolates the key homological criteria for verifying the existence of such properties. The Virasoro algebra is the universal central extension of the Witt algebra and plays a significant role in the definition of the vertex operator algebra. The theory of vertex operator algebras has provided a mathematical foundation for conformal field theory; see [Lepowsky 2005]. Understanding such field theories in two dimensions involves problems about the representation theory of the Virasoro and vertex operator algebras. The Witt algebra is an infinite-dimensional simple Lie algebra over C and is the smallest example of a Cartan-type Lie algebra. The Virasoro algebra has a triangular decomposition g = n− ⊕ h ⊕ n+ , which allows one to define a Category O. In this paper we study blocks of Category O for the Virasoro algebra, building on the foundational work of Fe˘ıgin and Fuchs [1990], who determined all maps between Verma modules for the Virasoro algebra. After making explicit the construction of BGG-resolutions for simple modules in these blocks (which is implicit in [Fe˘ıgin and Fuchs 1990]), we compute the n+ -cohomology with coefficients in any simple module. This extends results of Gonˇcarova [1973a; 1973b], who calculated H• (n+ , C), and of Rocha-Caridi and Wallach [1983a], who computed H• (n+ , L) for any simple module L in the trivial block. This cohomological information allows us to calculate the extensions between simple and Verma modules. We then verify that our categories satisfy properties given in [Cline et al. 1997]; in particular, they have a KL theory. These properties yield a computation of extensions between all simple modules and imply that truncated blocks of Category O for the Virasoro algebra give rise to interesting Koszul algebras. We find it quite remarkable that KL theories naturally arise in the representation theory of the Virasoro algebra. It would be interesting to determine if this occurs in a more general context within the representation theory of Cartan-type Lie algebras. The authors thank Brian Parshall for conversations about calculating extensions in quotient categories, Jonathan Kujawa for clarifying the connections between the extension theories used in Section 4.1, and the referee for several helpful comments and suggestions.

2. Notation and preliminaries 2.1. The Virasoro algebra is the Lie algebra g = C-span{z, dk | k ∈ Z} with bracket [ , ] given by [dk , z] = 0

and

[d j , dk ] = ( j − k)d j+k +

δ j,−k 3 ( j − j)z 12

for all j, k ∈ Z.

CATEGORY O FOR THE VIRASORO ALGEBRA: COHOMOLOGY AND KOSZULITY

3

The Virasoro algebra can be decomposed into a direct sum of subalgebras g = n− ⊕ h ⊕ n+ = n− ⊕ b+ , where n− = C-span{dn | n ∈ Z0 },

and b+ = h ⊕ n+ . There is an antiinvolution σ : g → g given by σ (dn ) = d−n and σ (z) = z. 2.2. Category O and other categories. The Category O consists of g-modules M such that L µ ∗ µ • M = µ∈h∗ M , where h = HomC (h, C) and M = {m ∈ M | hm = µ(h)m for all h ∈ h}; •

M is finitely generated as a g-module;

•

M is n+ -locally finite.

This definition is more restricted than the one given in [Moody and Pianzola 1995]. Identify each integer n ∈ Z with a weight n ∈ h∗ by n(d0 ) = n and n(z) = 0. Define a partial ordering on h∗ by (1)

µ < γ if µ = γ + n for some n ∈ Z>0 .

The category defined in [Moody and Pianzola 1995], which we denote O˜ , conL sists of g-modules M such that M = µ∈h∗ M µ , dim M µ < ∞, and there exist λ1 , . . . , λn ∈ h∗ such that M µ 6= 0 only for µ ≤ λi for some i. Then O (as we have defined it) is the full subcategory of O˜ consisting of finitely generated modules. Therefore, many of the results about O˜ proven in [Moody and Pianzola 1995] apply to O. For µ ∈ h∗ , the Verma module M(µ) is the induced module M(µ) = U (g) ⊗U (b+ ) Cµ . The Verma module M(µ) has a unique simple quotient, denoted L(µ). The modules L(µ) for µ ∈ h∗ provide a complete set of simple modules in Category O; see [Moody and Pianzola 1995, Section 2.3]. For µ, γ ∈ h∗ , define a partial ordering (2)

µγ

if L(µ) is a subquotient of M(γ ).

Extend this to an equivalence relation ∼. The blocks of g are the equivalence classes of h∗ determined by ∼. For each block [µ] ⊂ h∗ , define O[µ] to be the full subcategory of O such that, for any M ∈ O[µ] , the module L(γ ) is a subquotient L of M only for γ ∈ [µ]. For M ∈ O, M = [µ]⊂h∗ M [µ] , where M [µ] ∈ O[µ] ; see [Moody and Pianzola 1995, 2.12.4].

4


AND

EMILIE WIESNER

We will use several other categories. Let W be the category whose objects L are g-modules M such that M = λ∈h∗ M λ , where M λ is not necessarily finitedimensional. For a fixed weight µ ∈ h∗ , define W(µ) to be the full subcategory of L W whose objects are g-modules M such that M = λ≤µ M λ . The antiinvolution σ can be used to define a duality functor D on W. For L µ ∗ M ∈ W, define D M = µ (M ) (as a vector space) with g-action given by (x f )(v) = f (σ (x)v) for x ∈ g, f ∈ D M, and v ∈ M. Then, HomW (M, M 0 ) ∼ = 0 0 HomW (D M , D M) for all M, M ∈ W. Since σ (h) = h for h ∈ h, D M decomposes as a direct sum of weight spaces where (D M)µ = (M µ )∗ . Therefore, W(µ) is closed under D. Finally, note that DL ∼ = L for any simple module L ∈ O. 3. BGG resolutions and n+ -cohomology 3.1. Theorem 1 describes all Verma module embeddings in a given block of O. Since every nonzero map between Verma modules is an embedding, this describes all homomorphisms between Verma modules in a block. From this result one can construct BGG resolutions of the simple modules L(µ) to compute H• (n+ , L(µ)). Theorem 1 [Fe˘ıgin and Fuchs 1990, 1.9]. Suppose µ ∈ h∗ , and set h = µ(d0 ) and c = µ(z) ∈ C. Define √ p c − 13 + (c − 1)(c − 25) ν= and β = −4νh + (ν + 1)2 , 12 and consider the line in the r s-plane Lµ : r + νs + β = 0.

(3)

The Verma module embeddings involving M(µ) are determined by integer points (r, s) on Lµ : (i) Suppose Lµ passes through no integer points or one integer point (r, s) with r s = 0. Then the block [µ] is given by [µ] = {µ}. (ii) Suppose Lµ passes through exactly one integer point (r, s) with r s 6= 0. The block [µ] is given by [µ] = {µ, µ + r s}. The block structure is given below, where an arrow λ → γ between weights indicates M(λ) ⊆ M(γ ). µ = µ0

µ−1 = µ + rs

µ1 = µ + rs

µ = µ0

rs > 0

rs < 0


5

(r8 , s8 ) (r4 , s4 ) (r1 , s1 ) (r5 , s5 ) (r9 , s9 )

(r6 , s6 )

(r2 , s2 )

(r0 , s0 )

(r−1 , s−1 )

(r3 , s3 )

(r7 , s7 )

Figure 1. The points (r, s) on Lµ . (iii) Suppose Lµ passes through infinitely many integer points and crosses an axis at an integer point. Label these points (ri , si ) so that · · · < r−2 s−2 < r−1 s−1 < 0 = r0 s0 < r1 s1 < r2 s2 < · · · as in Figure 1. The block [µ] is given by [µ] = {µi = µ + ri si }. The block structure is given below. 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 11 00

00 11 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 00 11 00 11 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 0

1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 01 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 0

µ−1 µ = µ0 µ1

slope (Lµ ) > 0

µ−1 µ = µ0 µ1

1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 0

slope (Lµ ) < 0

(iv) Suppose Lµ passes through infinitely many integer points and does not cross either axis at an integer point. Label the integer points (ri , si ) on Lµ so that · · · < r−1 s−1 < r0 s0 < 0 < r1 s1 < r2 s2 < · · · . Also consider the auxiliary line e Lµ with the same slope as Lµ passing through the point (−r1 , s1 ). Label the integer points on this line (r 0j , s 0j ) in the same way as Lµ . The block [µ] is given by [µ] = {µi , µi0 }, where µi =

µ + ri si µ + r1 s1 + ri0 si0

for i odd, for i even,

The block structure is given below.

µi0 =

µ + ri+1 si+1 0 0 µ + r1 s1 + ri+1 si+1

for i odd, for i even.

6


µ′−2 µ′−1 µ′0 µ′1 µ′2

AND

EMILIE WIESNER

1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

µ′0

1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

µ′1

1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

µ′−2 µ′−1

µ−2 µ−1 µ = µ0

µ′2

µ−2

0 1 1 0 0 1 0 1 0 1 0 1 0 1 0 1

µ−1

0 1 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

µ = µ0 µ1

0 1 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1

µ2

µ1 µ2

slope (Lµ ) > 0

1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1

slope (Lµ ) < 0

We will refer to blocks as in case (iii) as “thin” blocks and blocks as in case (iv) as “thick” blocks. The second type of thick block has a highest weight poset structure equivalent to the Bruhat order on D∞ , the infinite dihedral group. 3.2. BGG resolutions. Fe˘ıgin and Fuchs [1988] observe without elaboration that their result, Theorem 1 above, yields a BGG resolution for the simple modules in Category O. We now provide the details for constructing these resolutions. Given a module M in Category O, define the radical of M, rad M, to be the smallest submodule such that M/ rad M is semisimple. Put rad0 M = M, and for i > 0 put radi M = rad(radi−1 M). This defines a decreasing filtration of M, the radical filtration. For i ≥ 0, layer i of the radical filtration is defined to be radi M = radi M/ radi+1 M. We also write hd M = M/ rad M. In general, see [Fe˘ıgin and Fuchs 1990], the terms and the layers of the radical filtration of M(µ), in the notation of Theorem 1(ii)–(iv), are as follows: (4)

rad0 M(µ) = L(µ),

radi M(µ) = M(µi ) + M(µi0 ) for i > 0, radi M(µ) = L(µi ) ⊕ L(µi0 )

for i > 0.

If µ belongs to a finite or thin block, then terms involving µi0 are to be ignored. Also if µ belongs to a block with a minimal element, say µn , then radn M(µ) = radn M(µ) = M(µn ) = L(µn ), radi M(µ) = radi M(µ) = 0 for i > n. Assume µ belongs to a thick block. According to [Bernˇste˘ın et al. 1975], there will be a complex C• → L(µ) → 0, where Ci is the direct sum of the Verma modules M(µi ) ⊕ M(µi0 ), provided that to each edge of the poset below µ, using the ordering , it is possible to assign a sign +1 or −1 so that the product of the signs on any diamond is −1. Such a labeling is indicated in Figure 2. It is easy to check directly in this case that the resulting complex is in fact a resolution, called a BGG resolution of L(µ).


µ

µ

+

+

+

µ′1 + −

µ2

+

+

µ3

+

µ′2

µ′n−2

+ −

−

µ2

+

+

µ3

+

+

µ′3

µ1

+

+ −

+

µ′1

µ1

µ′2

7

µ′3

+

µn−2 +

+

µ′n−1

±

±

µn−1 ∓

+

µn

Figure 2. Assignment of signs. Explicitly, the following are BGG resolutions of L(µ): for µ belonging to a thick block with a minimal element µn , 0 → M(µn ) → M(µn−1 ) ⊕ M(µ0n−1 ) → · · · → M(µ1 ) ⊕ M(µ01 ) → M(µ) → L(µ) → 0; and for µ belonging to a thick block with a maximal element, · · · → M(µi ) ⊕ M(µi0 ) → · · · → M(µ1 ) ⊕ M(µ01 ) → M(µ) → L(µ) → 0. Next consider a weight µ ∈ h∗ belonging to a thin block or a finite block. Then rad M(µ) = M(µ1 ) if µ1 exists in the block, and rad M(µ) = 0 otherwise. Thus, if M(µ) is not itself irreducible, the BGG resolution of L(µ) is 0 → M(µ1 ) → M(µ) → L(µ) → 0. We now introduce some additional notation. Fix µ ∈ h∗ . Define a length function l : [µ] → Z by l(µi ) = l(µi0 ) = i, using the notation of Theorem 1(ii)–(iv). While l( ) depends on a choice of representative µ for the block, the value (and, in particular, the parity) of l(ν) − l(γ ) for ν, γ ∈ [µ] is independent of the choice of representative. This will be relevant later.

8


AND

EMILIE WIESNER

In summary of the description given above, each simple module L(µ) has a BGG resolution · · · → C1 → C0 → L(µ) → 0 where L  Ll(ν)=i, νµ M(ν) if [µ] is a thick block or a finite block, (5) Ci = if i ≤ 1 and [µ] is a thin block, l(ν)=i M(ν)   0 if i > 1 and [µ] is a thin block. 3.3. n+ -cohomology. Gonˇcarova [1973a; 1973b] proved that (6)

Hk (n+ , C) = C(3k 2 +k)/2 ⊕ C(3k 2 −k)/2 .

Rocha-Caridi and Wallach use Gonˇcarova’s work to obtain BGG resolutions for C [1983a] and for the other simple modules in the trivial block [1983b]. Using this, they compute Hk (n+ , L) for any simple module L in the trivial block [1983b]. We extend their result to cohomology with coefficients in any simple module in O. Theorem 2. Let µ ∈ h∗ , and let k ∈ Z≥0 . (a) Suppose that µ belongs to a thick block or a finite block. As an h-module, M Hk (n+ , L(µ)) ∼ Cν . = Hk (n− , L(µ)) ∼ = ν ∈ [µ], ν µ l(ν) − l(µ) = k

(b) Suppose that µ belongs to a thin block. As an h-module,  M  Cν if k ≤ 1,   ν ∈ [µ], ν µ Hk (n+ , L(µ)) ∼ = Hk (n− , L(µ)) ∼ = l(ν) − l(µ) = k    0 if k > 1. Proof. We first compute the homology groups Hk (n− , L(µ)), where Hk (n− , −) is the k-th left derived functor of C ⊗U (n− ) −. Since Verma modules are free U (n− )modules, apply C ⊗U (n− ) − to the resolution (5). Note that C ⊗U (n− ) M(ν) ∼ = Cν . The resulting differential maps in the resolution are h-equivariant, and Cν appears at most once in the resolution for each weight ν. Therefore, all of the differential maps must be zero. This verifies that Hk (n− , L(µ)) is as stated. We now show that Hk (n− , L(µ)) ∼ = Hk (n+ , L(µ)). This may be well known; it is claimed in [Rocha-Caridi and Wallach 1983a] to follow from “standard arguments.” Because the infinite-dimensional case seems somewhat subtle, we include a proof for completeness. Write L = L(µ). Recall that Hk (n− , L) can be computed using the complex dk

· · · → 3k (n− ) ⊗ L → 3k−1 (n− ) ⊗ L → · · ·


9

and Hk (n+ , L) can be computed using the complex dk

· · · → 3k ((n+ )∗ ) ⊗ L → 3k+1 ((n+ )∗ ) ⊗ L → · · · . We extend the notion of duality defined in Section 2.2 to 3k (n− ), viewed as an h-module, as follows. For f ∈ ⊕λ∈h∗ ((3k (n− ))∗ )λ , define f˜ ∈ (3k (n+ ))∗ by f˜(x) = f (σ (x)) for x ∈ 3k (n+ ). Let D(3k (n− )) = { f˜ | f ∈ ⊕λ∈h∗ ((3k (n− ))∗ )λ }. Define 3k (D(n− )) ⊆ 3k ((n+ )∗ ) analogously. By choosing a basis for each weight space (3k (n− ) ⊗ L)λ , we can construct an h-module embedding 3k (n− ) ⊗ L → D(3k (n− )) ⊗ L ⊆ (3k (n+ ))∗ ⊗ L. Since D(L) ∼ = L, the differential map dk+1 induces a codifferential map d˜k : D(3k (n− ))⊗ L → D(3k+1 (n− ))⊗L as follows. Let f ∈ ⊕λ∈h∗ (3k (n− ))∗ )λ and g ∈ ⊕λ∈h∗ (L ∗ )λ . Then f ⊗g corresponds to an element ^ f ⊗ g ∈ D(3k (n− ))⊗DL ∼ = D(3k (n− ))⊗L. k+1 Define d˜k (^ f ⊗ g)(x ⊗m) = ( f ⊗ g)(dk+1 (σ (x)⊗m)) for x ∈ 3 (n+ ) and m ∈ L. For λ ∈ h∗ , it can be shown that dim(ker d˜k )λ − dim(Im d˜k−1 )λ = dim(ker dk )λ − dim(Im dk+1 )λ . This implies Hk (D(3• (n− )) ⊗ L) ∼ = Hk (n− , L). ∗ For each k, define φk = φ : 3k ((n+ )∗ ) → 3k (n+ ) by Y X sgn(τ ) φ( f 1 ∧ · · · ∧ f k )(x1 ∧ · · · ∧ xk ) = f i (xτ (i) )

(7)

τ ∈Sk

i

for xi ∈ n+ and f i ∈ (n+ )∗ . The map φ is an h-module isomorphism and satisfies φ(3k (D(n− ))) = D(3k (n− )). Moreover, it can be checked that (φk+1 ⊗ 1) ◦ d k = d˜k ◦ (φk ⊗ 1) on D(3k (n− )) ⊗ L. Therefore, φ ⊗ 1 gives an isomorphism (8)

Hk (D(3• (n− )) ⊗ L) ∼ = Hk (3• (D(n− )) ⊗ L).

To complete the proof, we need to show that Hk (3• (D(n− )) ⊗ L) ∼ = Hk (n+ , L), which entails checking that (i) for X ∈ Im(d k−1 ) ∩ 3k (D(n− )) ⊗ L, there is a Y ∈ 3k−1 (D(n− )) ⊗ L with d k−1 (Y ) = X ; (ii) for X ∈ ker(d k ), there is an X˜ ∈ 3k (D(n− )) ⊗ L such that X − X˜ ∈ Im(d k−1 ). Let X ∈ 3k ((n+ )∗ ) ⊗ L. The space 3k ((n+ )∗ ) ⊗ L decomposes as a direct Q product of its weight spaces. Write X = λ∈h∗ X λ , where X λ ∈ (3k ((n+ )∗ ) ⊗ L)λ . Then X λ = f 1 ⊗ m 1 + · · · + f n ⊗ m n for some f i ∈ 3k ((n+ )∗ )µi and m i ∈ L νi with µi + νi = λ. From the definition of 3k (D(n− )), we see that 3k (D(n− )) is the set Q of elements f ∈ 3k ((n+ )∗ ) with f = µ∈h∗ f µ such that f µ 6= 0 for only finitely many µ. Therefore, f i ∈ 3k (D(n− )), and so X λ ∈ 3k (D(n− )) ⊗ L. The differential map d k preserves weight spaces. Suppose X = d k−1 (Y˜ ) for Q some Y˜ ∈ 3k−1 ((n+ )∗ ) ⊗ L. Define Y = λ∈h∗ Yλ by Yλ = Y˜λ if X λ 6= 0 and

10


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Yλ = 0 otherwise. If X ∈ 3k (D(n− )) ⊗ L then Y ∈ 3k−1 (D(n− )) ⊗ L. This proves (i). Now let X ∈ ker(d k ). Then d k (X λ ) = 0 for all λ ∈ h∗ . Equations (7) and (8) imply that Hk (3• (D(n− )) ⊗ L) is finite dimensional. Because X λ ∈ 3k (D(n− )) ⊗ L, this means that there are only finitely many λ1 , . . . , λr such that X λi 6= 0 and X λi 6= d k (Yλi ) for some Yλi ∈ 3k−1 (D(n− ))⊗ L. Define X˜ λi = X λi for i = 1, . . . , r and X˜ λ = 0 otherwise. Then X˜ ∈ (3k (D(n− )) ⊗ L and X − X˜ ∈ Im(d k−1 ). This proves (ii). Corollary 3. Every irreducible module in Category O for the Virasoro algebra is a Kostant module (in the sense of [Boe and Hunziker 2006]). 4. Extensions The structure of the infinite blocks of O presents various obstacles in computing Ext-groups. The infinite blocks with a minimal element do not have enough projectives. In the infinite blocks with a maximal element, objects do not generally have finite length. We demonstrate that the first problem can be remedied by truncation, and the second can be addressed via a quotient construction. 4.1. Cohomology and truncated categories. We first define the truncation of an infinite block of O having a minimal element. Fix a weight µ in the block C, and consider the full subcategory C(µ) — called the truncation of the block at µ — of modules all of whose composition factors have highest weights less than or equal to µ, using the partial ordering given in (2). Now let C be a finite block of O, an infinite block of O with a maximal element, or a truncated infinite block with minimal element. Denote the weight poset of C by 3. Then there is a maximal element µ ∈ 3. If C is a truncated thick block, we can write µ = µ0 as in Theorem 1(iv). We assume that µ is chosen so that µ0 ≤ µ00 in the partial ordering given by Equation (1). Then 3 = {ν ∈ [µ] | ν ≤ µ}, which allows us to compare C and W(µ). We now use Theorem 2 to compute ExtiC (M(λ), L(ν)) by passing through relative cohomology and using the categories W and W(µ) defined in Section 2.2. Lemma 4. If λ, ν ∈ 3 and i ∈ Z≥0 , then ExtiW (M(λ), L(ν)) ∼ = Hi (n+ , L(ν))λ . Proof. For i ∈ Z≥0 , define Pi = U (g) ⊗U (h) 3i (g/h). Then, for any g-module M, the sequence (with suitably defined maps) · · · → P2 ⊗C M → P1 ⊗C M → M → 0 is a (g, h)-projective resolution [Kumar 2002, 3.1.8]. If M = M(λ), then


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Pi ⊗C M(λ) ∈ W, and so this gives a projective resolution in W. We then have ExtiW (M(λ), L(ν)) ∼ = Exti(g,h) (M(λ), L(ν)) ∼ = Exti(b+ ,h) (Cλ , L(ν)) ∼ = Exti + (C, C−λ ⊗ L(ν)) ∼ = Hi (b+ , h, C−λ ⊗ L(ν)) (b ,h) +

∼ = H (n , L(ν))λ . i

The second through fourth isomorphisms follow from [Kumar 2002, 3.1.14, 3.1.13, 3.9], respectively. The last isomorphism follows from definitions. There are two functors η : W → W(µ) and θ : W → W(µ) defined as follows. For M ∈ W, there is a unique minimal submodule M 0 ⊆ M such that M/M 0 ∈ W(µ). L Define ηM = M/M 0 . Note that M 0 is generated, as a g-module, by λ6≤µ M λ . Then, for any N , M ∈ W, and any g-module homomorphism f : M → N we have f (M 0 ) ⊆ N 0 . Therefore f induces a homomorphism from ηM to ηN . Define η( f ) to be this map. For M ∈ W, there is also a unique maximal submodule M 00 such that M 00 ∈ W(µ). Define θ M = M 00 . For any N , M ∈ W and any g-module homomorphism f : M → N , define θ(g) = g| M 00 . Using these functors we relate ExtiW (−, −) and ExtiW(µ) (−, −). Lemma 5. Let M, N ∈ W(µ). Then ExtiW(µ) (M, N ) = ExtiW (M, N ). Proof. First observe that η takes projectives to projectives and θ takes injectives to injectives. Let N → I• be an injective resolution in W. Since M ∈ W(µ), Homg (M, Ik ) ∼ = Homg (M, θ Ik ). Therefore, ExtiW (M, N ) = Hi (Homg (M, Ik )) ∼ = i H (Homg (M, θ Ik )). The lemma follows if we can show that θ is acyclic on N because this would imply that N = θ N → θ I• is an injective resolution. Note that Hi (θ I• ) = 0 if and only if Hi ((θ I• )γ ) = 0 for all γ ∈ h∗ . For γ ∈ h∗ , define Pγ = U (g) ⊗U (h) Cγ . Then (θ Ik )γ = Homg (Pγ , θ Ik ). This implies Hi ((θ I• )γ ) ∼ = Hi (Homg (Pγ , θ I• )) ∼ = Hi (Homg (η Pγ , I• )) ∼ = ExtiW (η Pγ , N ). Therefore, to complete the proof it is enough to show that ExtiW (η Pγ , N ) = 0 for i ≥ 1. There is a short exact sequence 0 → Pγ0 → Pγ → η Pγ → 0, which gives a long exact sequence 0 i i · · · → Exti−1 W (Pγ , N ) → ExtW (η Pγ , N ) → ExtW (Pγ , N ) → · · · . 0 Since Pγ is projective in W, ExtiW (Pγ , N ) = 0 for i ≥ 1. We claim Exti−1 W (Pγ , N ) = 0 for all i. To see this, let

Pγ6≤µ = span{m ∈ Pγ | m ∈ Pγν for some ν 6≤ µ, m = x ⊗ 1 for some x ∈ U (n+ )}. 6 ≤µ

Then Pγ

is a b+ -module.

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L Define Wb+ to be the category of b+ -modules M such that M = λ∈h∗ M λ . 6 ≤µ 6 ≤µ Define Q k = U (b+ ) ⊗U (h) 3k (b+ /h) ⊗C Pγ . Then Q • → Pγ is a projective 6 ≤µ resolution of Pγ in Wb+ ; see [Kumar 2002, 3.1.8]. Also, U (g)⊗U (b+ ) Q • is a pro6 ≤µ jective resolution of Pγ0 ∼ = U (g) ⊗U (b+ ) Pγ in W. Moreover, Homg (U (g) ⊗U (b+ ) Q k , N ) = Homb+ (Q k , N ) = 0 since Q νk 6= 0 only for ν 6≤ µ and N ν 6= 0 only for i 0 ν ≤ µ. Therefore Exti−1 W (Pγ , N ) = 0 for all i. This implies that ExtW (η Pγ , N ) = 0 for i ≥ 1. We now transfer the information from W(µ) to C. Theorem 6. Let C be a finite block of O, an infinite block of O with a maximal element, or a truncated infinite block with minimal element. Let 3 be the weight poset of C with maximal element µ, and let λ, ν ∈ 3. Then for i ≥ 0, (a) if C is a thick block or a finite block, ( C if λ ν and l(λ) − l(ν) = i, ExtiC (M(λ), L(ν)) ∼ = 0 otherwise; (b) if C is a thin block, ExtiC (M(λ),

( C L(ν)) ∼ = 0

if λ ν and l(λ) − l(ν) = i for i ≤ 1, otherwise.

Proof. Let M, N ∈ C. Given Theorem 2 and Lemmas 4 and 5, it is enough to show that ExtiC (M, N ) ∼ = ExtiW(µ) (M, N ). ∗ Let γ ∈ h , and recall that Pγ = U (g) ⊗U (h) Cγ . Then Pγ is projective in W, and thus η Pγ is projective in W(µ). Also, η Pγ is finitely generated: η Pγ is generated by 1 ⊗ 1 if γ ≤ µ and η Pγ = 0 otherwise. Thus, η Pγ ∈ O. Therefore, we Li can construct a resolution P• → M of M such that Pi = nj=1 η Pγ i ∈ O for some j γ ji ∈ h∗ , which is projective in W(µ). Recall that modules in O decompose according to blocks. Let P˜i be the component of Pi contained in C. (If C is a truncated block, the component of Pi corresponding to the full block will be contained in the truncation C since Pi ∈ W(µ) and C is truncated at µ.) Then, P˜• → M is a projective resolution in W(µ) and lies entirely in C. 4.2. Cohomology and quotient categories. Throughout this section, let C be an infinite block for the Category O with a maximal element. Then C is a highest weight category which contains enough projective objects. Let 3 be the corresponding weight poset indexing the simple objects in C. For λ ∈ 3 let P(λ) be the projective cover of L(λ). Set P = ⊕λ∈3 P(λ). Then P is a progenerator for C, and C is Morita equivalent to Mod(B), where B = EndC (P)op .


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We now apply results as described in [Cline et al. 1988, Theorem 3.5]. Let be a finite coideal, that is, = 3 − {γ ∈ 3 | γ δ} for some fixed δ ∈ 3. Consider P = ⊕λ∈ P(λ) and set A = EndC (P )op . Then there exists an idempotent e ∈ B, corresponding to the sum of identity maps in EndC (P(λ)) with λ ∈ such that eBe = A. Also, observe that the quotient category C() = Mod(A) is a highest weight category. For λ ∈ , set M (λ) = eM(λ), L (λ) = eL(λ), and P (λ) = e P(λ). Note that P (λ) is the projective cover of L (λ). The following proposition compares extensions between Verma modules and simple modules in C and C(). We remark that this result appears as [Cline et al. 2004, Corollary 3.5] with more finiteness restrictions. Proposition 7. Let λ, ν ∈ . For all i ≥ 0, ExtiC (M(λ), L(ν)) ∼ = ExtiC() (M (λ), L (ν)). Proof. Let λ, ν ∈ . According to [Doty et al. 2004, Theorem 2.2], there exists a first quadrant spectral sequence, i, j

i+ j

E 2 = ExtiB (Tor Aj (Be, M (λ)), L(ν)) ⇒ Ext A (M (λ), L (ν)), and by their [Theorem 4.5], Tor0A (Be, M (λ)) = M(λ). We need to show that Tor Aj (Be, M (λ)) = 0 for j ≥ 1. Then the spectral sequence above collapses and yields ∼ Exti (M (λ), L (ν)). Exti (M(λ), L(ν)) = B

A

for i ≥ 0 and λ, ν ∈ , as required. First we consider the case when j = 1. Since C() is a highest weight category, we may again invoke [Doty et al. 2004, Theorem 4.5] which states that Tor0A (Be, M (λ)) = M(λ) and M(λ) belongs to X; see their [Section 3.1] for a definition of X. Therefore, Tor1A (Be, M (λ)) = 0 by their [Proposition 3.1(A)]. To show that Tor Aj (Be, M (λ)) = 0 for j ≥ 2, we use induction on the ordering on the weights in . If λ is a maximal weight (relative to , the order introduced in (2)), then M (λ) is the projective cover of L (λ) and Tor Aj (Be, M (λ)) = 0 for j ≥ 1. Now suppose that Tor Aj (Be, M (µ)) = 0 for j ≥ 1 for all µ λ for µ ∈ . Consider the short exact sequence 0 → N → P (λ) → M (λ) → 0, where N has a filtration by modules M (µ) with µ λ. This induces a long exact sequence · · · ← Tor Aj−1 (Be, N ) ← Tor Aj (Be, M (λ)) ← Tor Aj (Be, P (λ)) ← . . .

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For j ≥ 1, Tor Aj (Be, P (λ)) = 0, and for j ≥ 2, Tor Aj−1 (Be, N ) = 0 by the induction hypothesis. Thus from the long exact sequence we can conclude for j ≥ 2 that Tor Aj (Be, M (λ)) = 0. Let L(λ) and L(ν) be simple B-modules with λ, ν ∈ . Then eL(λ) 6= 0 and eL(ν) 6= 0. Let · · · → P2 → P1 → P0 → L(λ) → 0 be the minimal projective resolution of L(λ) in C. Set n+1 (L(λ)) to be the kernel of the map Pn → Pn−1 . By convention, we let 0 (L(λ)) = L(λ). Under a suitable condition on the minimal projective resolution, we can compare extensions between simple modules in C and C(). This comparison depends on bounding the composition factors in the projective resolution of L(λ). The following proposition provides such a bound. Proposition 8. Let λ, ν ∈ 3. (i) If ExtnC (rad M(λ), L(ν)) 6= 0 then l(λ) − l(ν) ≤ n − 1. (ii) If ExtnC (L(λ), L(ν)) 6= 0 then l(λ) − l(ν) ≤ n. Proof. In this proof we assume that C is a thick block. In the case that C is a thin block or a finite block, the proposition follows from similar arguments. (i) We prove this by induction on n. Let n = 0. Since rad M(λ) = M(λ1 )+M(λ01 ), with notation as in Theorem 1(iv), HomC (rad M(λ), L(ν)) 6= 0 if and only if ν = λ1 or λ01 , whence l(λ) − l(ν) = −1. Assume the result is true for n −1 and all pairs of weights in 3. Recall from (4) that rad M(λ) = M(λ1 ) + M(λ01 ) and rad2 M(λ) = M(λ2 ) + M(λ02 ) = rad M(λ1 ) = rad M(λ01 ). Thus we have a short exact sequence 0 → rad M(λ1 ) = rad M(λ01 ) → M(λ1 ) ⊕ M(λ01 ) → rad M(λ) → 0, where the inclusion sends x to (x, −x) and the surjection sends (x, y) to x + y. This induces a long exact sequence n · · · → Extn−1 C (rad M(λ1 ), L(ν)) → ExtC (rad M(λ), L(ν))

→ ExtnC (M(λ1 ) ⊕ M(λ01 ), L(ν)) → · · · . Suppose l(λ) − l(ν) > n − 1. Then l(λ1 ) − l(ν) = l(λ) + 1 − l(ν) > n. This implies ExtnC (M(λ1 ), L(ν)) = 0 by Theorem 6, and similarly for λ01 . Also, l(λ1 ) − n−1 l(ν) > (n − 1) + 1, so ExtC (rad M(λ1 ), L(ν)) = 0 by induction. This implies n ExtC (rad M(λ), L(ν)) = 0. (ii) The proof is again by induction on n. The result is clear for n = 0. Assume it is true for n − 1. Consider the short exact sequence 0 → rad M(λ) → M(λ) → L(λ) → 0.


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This induces a long exact sequence n n · · · → Extn−1 C (rad M(λ), L(ν)) → ExtC (L(λ), L(ν)) → ExtC (M(λ), L(ν)) → · · · .

Suppose l(λ) − l(ν) > n. Then Extn−1 C (rad M(λ), L(ν)) = 0 by part (i), and n ExtC (M(λ), L(µ)) = 0 by Theorem 6. This implies ExtnC (L(λ), L(ν)) = 0. Recall P• → L(λ) is a minimal projective resolution of L(λ). For γ ∈ 3, if L(γ ) is j a composition factor of hd P j , then ExtC (L(λ), L(γ )) 6= 0. Therefore, Proposition 8 gives a bound on the composition factors which can appear in hd P j . This is the condition needed to compare extensions between simple modules in C and C(). Proposition 9. Let λ, ν ∈ and define N = min{|l(λ) − l(γ )| : γ ∈ 3 − } − 1. Then, for j = 0, 1, . . . , N , j j ExtC (L(λ), L(ν)) ∼ = ExtC() (L (λ), L (ν)).

Proof. We first claim that Be P j = P j for j = 0, . . . , N . Note that Be P j = P j if and only if hd P j contains no composition factors that are killed by the idempotent j e. Suppose that L(γ ) ⊆ hd P j . Then ExtC (L(γ ), L(λ)) 6= 0. From the proof of Theorem 6 and using the duality on W(µ), we have j j ExtC (L(γ ), L(λ)) ∼ = ExtW(µ) (L(γ ), L(λ)) j j ∼ = ExtW(µ) (L(λ), L(γ )) ∼ = ExtC (L(λ), L(γ )).

Then Proposition 8 implies that |l(λ) − l(γ )| ≤ j. Therefore, for j ≤ N , γ ∈ , and so eL(γ ) 6= 0. Since hd j (L(λ)) ∼ = hd P j , we have Be j (L(λ)) = j (L(λ)) for j = 0, 1, 2, . . . , N . Let j = 0, 1, . . . , N − 1. Since Be j (L(λ)) = j (L(λ)) there exists a short exact sequence [Doty et al. 2004, Theorem 3.2], 0 → j+1 (L(λ))/Be j+1 (L(λ)) → Tor0A (Be, e j (L(λ))) → j (L(λ)) → 0. Note that we are using that 1 ( j (L(λ))) ∼ = j+1 (L(λ)). Since j+1 (L(λ)) = j+1 Be (L(λ)), we have Tor0A (Be, e j (L(λ))) ∼ = j (L(λ)).

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Finally, we can apply [Doty et al. 2004, Theorem 2.4(B)(ii)] and a dimension shifting argument twice (see [Benson 1998, Corollary 2.5.4]) to see that j+1

Ext B (L(λ), L(ν)) ∼ = Ext1B ( j (L(λ)), L(ν)) ∼ Ext1 (Tor A (Be, e j (L(λ))), L(ν)) = B 0 ∼ Ext1 (e j (L(λ)), eL(ν)) = A 1 j ∼ Ext = A ( (eL(λ)), eL(ν)) j+1 ∼ = Ext (eL(λ), eL(ν)) A

j+1 ∼ = Ext A (L (λ), L (ν)).

To justify the step between lines 3 and 4 in the identifications above, observe that the idempotent functor e(−) : Mod(B) → Mod(A) is exact. Moreover, Be P j = P j for j = 1, 2, . . . , N so we have an exact sequence of projective A-modules: e PN → e PN −1 → . . . e P1 → e P0 → eL(λ) → 0. This implies that e j (L(λ)) ∼ = j (eL(λ)) ⊕ Q j , where Q j is a projective Amodule for j = 0, 1, . . . , N . 4.3. Extensions between simple modules. Let C be a finite block, an infinite block with a maximal element, or a truncation of an infinite block with a minimal element. Let 3 be the weight poset of C, with length function l : 3 → Z. Theorem 10. Let λ, ν ∈ 3. Then, (a) if C is a thick block or a finite block, dim ExtnC (L(λ), L(ν)) = #{γ ∈ 3 | γ λ, ν; 2l(γ ) − l(λ) − l(ν) = n}; (b) if C is a thin block, dim ExtnC (L(λ), L(ν)) = #{γ ∈ 3 | γ λ, ν; 2l(γ ) − l(λ) − l(ν) = n} if n ≤ 2 and equals zero otherwise. In particular, ExtnC (L(λ), L(ν)) 6= 0 only when n ≡ (l(λ) − l(ν)) (mod 2). Proof. Suppose C is an infinite block with a maximal element. Let be a finite coideal of 3 containing λ, ν. From Proposition 7, we know ExtiC (M(λ), L(ν)) = ExtiC() (M (λ), L (ν)). For a fixed n ∈ Z>0 , we assume that is sufficiently large so that γ ∈ for all γ ∈ 3 with |l(λ)−l(γ )| ≤ n. Then Proposition 9 implies that ExtnC (L(λ), L(ν)) ∼ = ExtnC() (L (λ), L (ν)). Thus, by replacing C by a quotient category C() where appropriate, we may assume that C is a highest weight category with finite weight poset 3. Because the objects of C have finite composition length, C is closed under the duality D on W(µ). Define A(γ ) = D M(γ ).


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Now apply [Cline et al. 1993, 3.5]: X j dim ExtnC (L(λ), L(ν)) = dim ExtiC (L(λ), A(γ )) dim ExtC (M(γ ), L(ν)). γ ∈3, i, j∈Z≥0 i+ j=n

Using the duality on C, ExtiC (L(λ), A(γ )) ∼ = ExtiC (M(γ ), L(λ)). Then Theorem 6 gives the result. 4.4. Ext1 -quivers. Let C be a finite block, a quotient of an infinite block with a maximal element, or a truncation of an infinite block with a minimal element. Let 3 be the (finite) weight poset of C. The Ext1 -quiver of C is defined to be the directed graph with vertices labelled by 3 and with dim Ext1C (L(λ), L(µ)) edges from λ to µ. It is clear from Theorem 10 and Proposition 9 that the Ext1 -quiver of C is obtained from the poset 3 simply by replacing each edge by a pair of directed edges, one pointing in each direction. The edges from λ to µ can also be viewed as representing linearly independent elements of HomC (P(λ), P(µ)) in the finite dimensional algebra op M A = EndC P(λ) . λ∈3

One can ask for the relations that exist between the maps in this algebra, which provides a presentation of the algebra by the quiver with relations. Suppose that C is either a finite block or a finite quotient or truncation of a thin block. Then 3 is a simple chain, say of length n, and it is quite easy to write down the structure of the projective indecomposables P(λ). This is done for n = 4 in [Futorny et al. 2001], and the pattern is the same for any n. Moreover, if the elements of 3 are numbered λ1 , . . . , λn from top to bottom, and if αi and βi represent the maps from P(λi ) to P(λi+1 ) and from P(λi+1 ) to P(λi ), respectively, for 1 ≤ i ≤ n − 1, then one sees easily that, up to scalar multiples, (9)

α1 β1 = 0

and βi αi = αi+1 βi+1

for 1 ≤ i ≤ n − 2

Note that maps compose left-to-right, because of the ()op in the definition of A. Now we can assume we’re working in the basic algebra with simple modules (respectively, projective indecomposable modules) labelled by b L(λi ) (respectively, b i )) for i = 1, 2, . . . , n. Note that dim b P(λ L(λi ) = 1 for every i so that dim A = Pn b b i=1 dim P(λi ), which is easy to compute using the known structures of the P(λi ). On the other hand, using the relations given in (9), one can check directly that there Pn b i ) linearly independent words in the αi and βi . Thus (9) are at most i=1 dim P(λ must be all the relations. In contrast, suppose that C is a finite quotient or truncation of a thick block. Then the poset 3 is isomorphic to the Bruhat order on a dihedral group. In this

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case the structure of the projectives, and the exact nature of the relations, seem to be quite difficult to deduce. For example, Stroppel [2003] works out the relations for the Ext1 -quiver of the regular blocks of Category O for the finite simple complex rank 2 Lie algebras, using some deep results of Soergel. Not only are the answers quite complicated (for example, for G 2 there are 70 relations), but as far as we are aware the analogs of Soergel’s results are not known for the Virasoro algebra. Nonetheless, based on Stroppel’s computations, we speculate that the relations in the Ext1 -quiver of C are all quadratic. 5. Kazhdan–Lusztig theories and Koszulity 5.1. Let B = B0 ⊕ B1 ⊕ · · · ⊕ Bq be a finite-dimensional graded algebra, and let gr C B be the category of finite-dimensional graded B-modules. Regard every simple gr B-module L as concentrated in degree zero; then the simple modules in C B can be obtained by shifting the gradings of the simple B-modules. If L is a simple gr B-module then L(i) will denote the simple module in C B by shifting i places to the right; see [Cline et al. 1997, Section 1.3]. The algebra B is Koszul if for all simple B-modules L and L 0 and m, n, p ∈ Z, (10)

p

ExtCgr (L(m), L 0 (n)) 6= 0 ⇒ n − m = p. B

Now let C be either a finite block of O, a truncation of an infinite block of O with a minimal element, or a quotient of an infinite block of O with a maximal element. Then C is a highest weight category (with duality) having a finite weight poset 3 and length function l. Moreover, Theorem 6 implies that C has a Kazhdan–Luzstig theory, as defined in [Cline et al. 1993]. Recall that C is equivalent to Mod(A) for a finite-dimensional algebra A. Let gr A be the associated graded algebra obtained L by using the radical filtration on A. Moreover, set L = λ∈3 L(λ), and define the homological dual of A to be A! = Ext•C (L , L). The following theorem establishes Koszulity results on A. Theorem 11. Let C be as described above, and let A be the associated quasihereditary algebra. Then (a) A! is Koszul, (b) gr A is Koszul, and (c) (A! )! ∼ = gr A. Proof. To prove the theorem, it suffices to check the condition (SKL0 ) ExtnC (radi M(λ), L(µ)) 6= 0 ⇒ n ≡ l(µ) − l(λ) + i (mod 2) for all λ, µ ∈ 3. In principle, that the same parity vanishes for ExtnC (L(λ), A(µ)/soci A(µ)) should also be checked, but this follows by duality in our setting.


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gr

Once (SKL0 ) is established, then by [Cline et al. 1997, Lemma 2.1.5] C A! has a gr graded KL-theory, and so C A! and A! are Koszul using [Cline et al. 1994, Theorem 3.9]. The condition (SKL0 ) implies parts (b) and (c) by [Cline et al. 1997, Theorem 2.2.1]. The case i = 0 of (SKL0 ) follows immediately from Theorem 6. Assume i > 0. Then radi M(λ) is either 0, a Verma module M(ν) with l(ν) − l(λ) = i, or a sum of two such Verma modules. The first case is trivial, and in the second, we can use the same argument as for i = 0. So assume we are in the third case. Let λi and λi0 be the two elements ν satisfying l(ν) − l(λ) = i. We have a short exact sequence 0 → M(λi ) → radi M(λ) → L(λi0 ) → 0. The corresponding long exact sequence is · · · → ExtnC (L(λi0 ), L(µ)) → ExtnC (radi M(λ), L(µ)) → ExtnC (M(λi ), L(µ)) → · · · , and we are assuming the middle term is nonzero. Then one of the two adjacent terms must be nonzero. If the term on the right is nonzero, then the same argument as for i = 0 gives the desired parity condition, since l(λi ) = l(λ) + i. If the term on the left is nonzero, then by Theorem 10 we have n ≡ l(λi0 ) −l(µ) ≡ l(λ) + i −l(µ) (mod 2), which is equivalent to the required condition. Remarks. 1. Since this proof holds for all quotient categories of C, this shows that C satisfies the strong Kazhdan Lusztig condition; see [Cline et al. 1997, 2.4.1]. 2. Suppose C comes from either a finite or a thin block of O. Since the relations in the Ext1 -quiver of C are all homogeneous (in fact quadratic; see (9)), it follows that A itself is tightly graded (that is, A ∼ = gr A). In particular in this case A is Koszul. 3. An interesting open question is to determine whether A is tightly graded or whether A itself is Koszul when A is associated to a thick block. The answers would be affirmative if the relations are all quadratic, as speculated in Section 4.4. References [Be˘ılinson and Bernstein 1981] A. Be˘ılinson and J. Bernstein, “Localisation de g-modules”, C. R. Acad. Sci. Paris Sér. I Math. 292:1 (1981), 15–18. MR 82k:14015 Zbl 0476.14019 [Beilinson et al. 1996] A. Beilinson, V. Ginzburg, and W. Soergel, “Koszul duality patterns in representation theory”, J. Amer. Math. Soc. 9:2 (1996), 473–527. MR 96k:17010 Zbl 0864.17006 [Benson 1998] D. J. Benson, Representations and cohomology. I, Second ed., Cambridge Studies in Advanced Mathematics 30, Cambridge University Press, Cambridge, 1998. MR 99f:20001a Zbl 0908.20001

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[Bernšte˘ın et al. 1975] I. N. Bernšte˘ın, I. M. Gel0 fand, and S. I. Gel0 fand, “Differential operators on the base affine space and a study of g-modules”, pp. 21–64 in Lie groups and their representations (Budapest, 1971), edited by I. M. Gel0 fand, Halsted, New York, 1975. MR 58 #28285 Zbl 0338.58019 [Bernšte˘ın et al. 1976] I. N. Bernšte˘ın, I. M. Gel0 fand, and S. I. Gel0 fand, “A certain category of g-modules”, Funct. Anal. Appl. 10 (1976), 87–92. Zbl 0353.18013 [Boe and Hunziker 2006] B. D. Boe and M. Hunziker, “Kostant modules in blocks of category O S ”, Preprint, 2006. arXiv math.RT/0604336 [Brylinski and Kashiwara 1981] J.-L. Brylinski and M. Kashiwara, “Kazhdan–Lusztig conjecture and holonomic systems”, Invent. Math. 64:3 (1981), 387–410. MR 83e:22020 Zbl 0473.22009 [Cline et al. 1988] E. Cline, B. Parshall, and L. Scott, “Finite-dimensional algebras and highest weight categories”, J. Reine Angew. Math. 391 (1988), 85–99. MR 90d:18005 Zbl 0657.18005 [Cline et al. 1993] E. Cline, B. Parshall, and L. Scott, “Abstract Kazhdan–Lusztig theories”, Tohoku Math. J. (2) 45:4 (1993), 511–534. MR 94k:20079 Zbl 0801.20013 [Cline et al. 1994] E. Cline, B. Parshall, and L. Scott, “The homological dual of a highest weight category”, Proc. London Math. Soc. (3) 68:2 (1994), 294–316. MR 94m:20093 Zbl 0819.20045 [Cline et al. 1997] E. Cline, B. Parshall, and L. Scott, “Graded and non-graded Kazhdan-Lusztig theories”, pp. 105–125 in Algebraic groups and Lie groups, edited by G. Lehrer et al., Austral. Math. Soc. Lect. Ser. 9, Cambridge Univ. Press, Cambridge, 1997. MR 99h:16015 Zbl 0880.16004 [Cline et al. 2004] E. Cline, B. Parshall, and L. Scott, “On Ext-transfer for algebraic groups”, Transform. Groups 9:3 (2004), 213–236. MR 2005e:20069 Zbl 1063.20049 [Doty et al. 2004] S. R. Doty, K. Erdmann, and D. K. Nakano, “Extensions of modules over Schur algebras, symmetric groups and Hecke algebras”, Algebr. Represent. Theory 7:1 (2004), 67–100. MR 2005e:20004 Zbl 1084.20004 [Fe˘ıgin and Fuchs 1988] B. L. Fe˘ıgin and D. B. Fuchs, “Cohomology of some nilpotent subalgebras of the Virasoro and Kac–Moody Lie algebras”, J. Geom. Phys. 5:2 (1988), 209–235. MR 91c:17012 Zbl 0692.17008 [Fe˘ıgin and Fuchs 1990] B. L. Fe˘ıgin and D. B. Fuchs, “Representations of the Virasoro algebra”, pp. 465–554 in Representation of Lie groups and related topics, edited by A. M. Vershik and D. P. Zhelobenko, Adv. Stud. Contemp. Math. 7, Gordon and Breach, New York, 1990. MR 92f:17034 Zbl 0722.17020 [Futorny et al. 2001] V. Futorny, D. K. Nakano, and R. D. Pollack, “Representation type of the blocks of category O”, Q. J. Math. 52:3 (2001), 285–305. MR 2002h:17006 Zbl 1021.17008 [Gonˇcarova 1973a] L. V. Gonˇcarova, “Cohomology of Lie algebras of formal vector fields on the line”, Funkcional. Anal. i Priložen. 7:2 (1973), 6–14. In Russian; translated in Funct. Anal. Appl. 7 (1973), 91–97. MR 49 #4058a Zbl 0284.17006 [Gonˇcarova 1973b] L. V. Gonˇcarova, “Cohomology of Lie algebras of formal vector fields on the line”, Funkcional. Anal. i Priložen. 7:3 (1973), 33–44. MR 49 #4058b Zbl 0289.57026 [Irving 1990] R. S. Irving, “BGG algebras and the BGG reciprocity principle”, J. Algebra 135:2 (1990), 363–380. MR 92d:17011 Zbl 0725.17007 [Irving 1992] R. S. Irving, “Graded BGG algebras”, pp. 181–200 in Abelian groups and noncommutative rings, Contemp. Math. 130, Amer. Math. Soc., Providence, RI, 1992. MR 94e:16013 [Kazhdan and Lusztig 1979] D. Kazhdan and G. Lusztig, “Representations of Coxeter groups and Hecke algebras”, Invent. Math. 53:2 (1979), 165–184. MR 81j:20066 Zbl 0499.20035


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[Kumar 2002] S. Kumar, Kac–Moody groups, their flag varieties and representation theory, Progress in Mathematics 204, Birkhäuser, Boston, 2002. MR 2003k:22022 Zbl 1026.17030 [Lepowsky 2005] J. Lepowsky, “From the representation theory of vertex operator algebras to modular tensor categories in conformal field theory”, Proc. Natl. Acad. Sci. USA 102:15 (2005), 5304– 5305. MR 2006a:17027 Zbl 1112.17030 [Moody and Pianzola 1995] R. V. Moody and A. Pianzola, Lie algebras with triangular decompositions, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, New York, 1995. MR 96d:17025 Zbl 0874.17026 [Rocha-Caridi and Wallach 1983a] A. Rocha-Caridi and N. R. Wallach, “Characters of irreducible representations of the Lie algebra of vector fields on the circle”, Invent. Math. 72:1 (1983), 57–75. MR 85a:17010 Zbl 0498.17010 [Rocha-Caridi and Wallach 1983b] A. Rocha-Caridi and N. R. Wallach, “Highest weight modules over graded Lie algebras: resolutions, filtrations and character formulas”, Trans. Amer. Math. Soc. 277:1 (1983), 133–162. MR 84m:17005 Zbl 0512.17007 [Stroppel 2003] C. Stroppel, “Category O: quivers and endomorphism rings of projectives”, Represent. Theory 7 (2003), 322–345. MR 2004h:17007 Zbl 1050.17005 Received January 19, 2007. Revised July 23, 2007. B RIAN D. B OE D EPARTMENT OF M ATHEMATICS U NIVERSITY OF G EORGIA ATHENS , GA 30602 [email protected] http://www.math.uga.edu/~brian/ DANIEL K. NAKANO D EPARTMENT OF M ATHEMATICS U NIVERSITY OF G EORGIA ATHENS , GA 30602 [email protected] http://www.math.uga.edu/~nakano/ E MILIE W IESNER M ATHEMATICS D EPARTMENT W ILLIAMS H ALL 212, I THACA C OLLEGE 953 DANBY ROAD I THACA , NY 14850 [email protected] http://faculty.ithaca.edu/ewiesner/


CONVEXITY IN LOCALLY CONFORMALLY FLAT MANIFOLDS WITH BOUNDARY M ARCOS P ETRÚCIO DE A. C AVALCANTE Given a closed subset 3 of the open unit ball B1 ⊂ Rn for n ≥ 3, we consider a complete Riemannian metric g on B 1 \3 of constant scalar curvature equal to n(n − 1) and conformally related to the Euclidean metric. We prove that every closed Euclidean ball B ⊂ B1 \ 3 is convex with respect to the metric g, assuming the mean curvature of the boundary ∂ B1 is nonnegative with respect to the inward normal.

1. Introduction Let B1 denote the open unit ball of Rn for n ≥ 3. Given a closed subset 3 ⊂ B1 , we will consider a complete Riemannian metric g on B 1 \ 3 of constant positive scalar curvature R(g) = n(n − 1) and conformally related to the Euclidean metric δ. We will also assume that g has nonnegative boundary mean curvature. Here and throughout, second fundamental forms will be computed with respect to the inward unit normal vector. In this paper we prove Theorem 1.1. If B ⊂ B1 \ 3 is a standard Euclidean ball, then ∂ B is convex with respect to the metric g. Here, we say that ∂ B is convex if its second fundamental form is positive definite. Since ∂ B is umbilical in the Euclidean metric and the notion of an umbilical point is conformally invariant, we know that ∂ B is also umbilic in the metric g. In that case, ∂ B is convex if its mean curvature h is positive everywhere. This theorem is motivated by an analogous one on the sphere due to R. Schoen [1991]. He shows that if 3 ⊂ S n for n ≥ 3 is closed and nonempty and g is a complete Riemannian metric on S n \ 3 that is conformal to the standard round metric g0 and has constant positive scalar curvature n(n − 1), then every standard ball B ⊂ S n \3 is convex with respect to the metric g. Schoen used this geometrical result to prove the compactness of the set of solutions to the Yamabe problem in the locally conformally flat case. D. Pollack [1993] also used Schoen’s theorem to MSC2000: primary 53A30, 53C21; secondary 52A20. Keywords: scalar curvature, locally conformally flat metric, convexity. 23

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MARCOS PETRÚCIO DE A. CAVALCANTE

prove a compactness result for the singular Yamabe problem on the sphere where the singular set is a finite collection of points 3 = { p1 , . . . , pk } ⊂ S n for n ≥ 3. In this context Theorem 1.1 can be viewed as the first step in the direction of proving compactness for the singular Yamabe problem with boundary conditions. As we will see, the problem of finding a metric satisfying the hypotheses of Theorem 1.1 is equivalent to finding a positive solution to an elliptic PDE with critical Sobolev exponent. This problem is invariant by conformal transformations. So, by applying a convenient inversion on the Euclidean space, we may consider the same problem on an unbounded subset of Rn . The idea of the proof is to show that if ∂ B is not convex, we can find a smaller ball e B ⊂ B with a nonconvex boundary as well. To do this we will use the hypothesis on the mean curvature of ∂ B1 and get geometrical information from that equation by applying the moving planes method as in [Gidas et al. 1979]. The contradiction follows by the construction of these balls. 2. Preliminaries Here we will introduce some notation and recall some results that will be used in the proof of Theorem 1.1. We will also describe a useful example. Let (M n , g0 ) for n ≥ 3 be a smooth orientable Riemannian manifold, possibly with boundary. Let us denote by R(g0 ) its scalar curvature and by h(g0 ) its boundary mean curvature. Let g = u 4/(n−2) g0 be a metric conformal to g0 . Then the positive function u satisfies the following nonlinear elliptic partial differential equation with critical Sobolev exponent: 1g0 u − (1)

n −2 n −2 R(g0 )u + R(g)u (n+2)/(n−2) = 0 in M, 4(n −1) 4(n −1) ∂u n −2 n −2 − h(g0 )u + h(g)u n/(n−2) = 0 on ∂ M, ∂ν 2 2

where ν is the inward unit normal vector field to ∂ M. The problem of existence of solutions to (1) when R(g) and h(g) are constants is referred to as the Yamabe problem. It was completely solved when ∂ M = ∅ in a sequence of works, beginning with H. Yamabe himself [1960], followed by N. Trudinger [1968] and T. Aubin [1976], and finally by R. Schoen [1984]. In the case of nonempty boundary, J. Escobar solved almost all the cases [1992a; 1992b], followed by Z. Han and Y. Li [1999], F. Marques [2005], and others. Here, however, we wish to study solutions of (1) with R(g) constant; these become singular on a closed subset 3 ⊂ M. This is the so called singular Yamabe problem. This singular behavior is equivalent, at least in the case that g0 is conformally flat, to requiring g to be complete on M \ 3. The existence problem (with ∂ M = ∅) displays a relationship between the size of 3 and the sign of R(g). It

CONVEXITY IN LOCALLY CONFORMALLY FLAT MANIFOLDS WITH BOUNDARY

25

is known that for a solution with R(g) < 0 to exist, it is necessary and sufficient that dim(3) > (n −2)/2 (see [Aviles and McOwen 1988; McOwen 1993; Finn and McOwen 1993]), while if a solution exists with R(g) ≥ 0, then dim(3) ≤ (n−2)/2. Here dim(3) stands for the Hausdorff dimension of 3. In this paper we will treat the case of constant positive scalar curvature, which we suppose equal to n(n − 1) after normalization. In this case the simplest examples are given by the Fowler solutions which we will now discuss briefly. Let u : Rn \ {0} → R be a positive smooth function such that (2)

n(n −2) (n+2)/(n−2) u = 0 in Rn \ {0} for n ≥ 3 and 4 0 is an isolated singularity.

1u +

Then g = u 4/(n−2) δ is a complete metric on Rn \ {0} of constant scalar curvature n(n − 1). Using the invariance under conformal transformations we may work in different background metrics. The most convenient one here is the cylindrical metric gcyl = dθ 2 + dt 2 on S n−1 × R. Then g = v 4/(n−2) gcyl , where v is defined in the whole cylinder and satisfies (3)

d 2v (n −2)2 n(n −2) (n+2)/(n−2) + 1θ v − v+ v = 0. 2 4 4 dt

One easily verifies that the solutions to Equation (2) and (3) are related by (4)

u(x) = |x|(2−n)/2 v(x/|x|, − log |x|).

By a deep theorem of Caffarelli, Gidas and Spruck [1989, Theorem 8.1], we know that v is rotationally symmetric, that is v(θ, t) = v(t), and therefore the PDE (3) reduces to the ODE d 2 v (n −2)2 n(n −2) (n+2)/(n−2) − v+ v = 0. 4 4 dt 2 Setting w = v 0 this equation is transformed into a first order Hamiltonian system dv = w, dt dw (n −2)2 n(n −2) (n+2)/(n−2) = v− v , dt 4 4 whose Hamiltonian energy is H (v, w) = w2 −

(n −2)2 2 (n −2)2 2n/(n−2) v + v . 4 4

The solutions (v(t), v 0 (t)) describe the level sets of H , and we note that (0, 0) and (±v0 , 0), where v0 = ((n −2)/n)(n−2)/4 , are the equilibrium points. We restrict

26


ourselves to the half-plane {v > 0} where g = v 4/(n−2) gcyl has geometrical meaning. On the other hand, we are looking for complete metrics. Those will be generated by the Fowler solutions, that is, the periodic solutions around the equilibrium point (v0 , 0). They are symmetric with respect to v-axis and can be parametrized by the minimum value ε attained by v for ε ∈ (0, v0 ] (and a translation parameter T ). We will denote them by vε . We point out that v0 corresponds to the scaling of gcyl that makes the cylinder S n−1 × R have scalar curvature n(n − 1). One obtains the Fowler solutions u ε in Rn \ {0} by using the relation (4). We can now construct metrics satisfying the hypotheses of Theorem 1.1 (with 3 = {0}) from the Fowler solutions. To do this, we just take a Fowler solution v defined for t ≥ t0 , where t0 is such that w = dv/dt ≤ 0 or equivalently h(g) = −

2 −n/(n−2) dv v ≥ 0. n −2 dt

We point out that, by another result of Caffarelli, Gidas, and Spruck [1989, Theorem 1.2], it is known that, given a positive solution u to 1u +

(5)

n(n −2) (n+2)/(n−2) u =0 4

that is defined in the punctured ball B1 \ {0} and that is singular at the origin, there exists a unique Fowler solution u ε such that u(x) = (1 + o(1))u ε (|x|) as |x| → 0. Therefore, from Equation (4) or also [Korevaar et al. 1999], either u extends as a smooth solution to the ball, or there exist positive constants C1 and C2 such that C1 |x|(2−n)/2 ≤ u(x) ≤ C2 |x|(2−n)/2 . 3. Proof of Theorem 1.1 The proof will be by contradiction. If ∂ B is not convex then, since it is umbilical, there exists a point q ∈ ∂ B such that the mean curvature of ∂ B at q (with respect to the inward unit normal vector) is H (q) ≤ 0. If we write g = u 4/(n−2) δ, then u is a positive smooth function on B 1 \ 3 satisfying n(n −2) (n+2)/(n−2) u =0 4 ∂u n −2 n −2 n/(n−2) − u+ hu =0 ∂ν 2 2

1u + (6)

in B1 \ 3, on ∂ B1 .

Now, we will choose a point p ∈ ∂ B with p 6= q and consider the inversion I : Rn \ { p} → Rn \ { p}.


27

This map takes B 1 \ ({ p} ∪ 3) on Rn \ (B(a, r ) ∪ 3), where B(a, r ) is an open ball of center a ∈ Rn and radius r > 0 and 3 still denotes the singular set. Let us denote by 6 the boundary of B(a, r ), that is, 6 = I (∂ B1 ). The image of ∂ B \ { p} is a hyperplane 5, and by a coordinate choice we may assume 5 = 50 := {x ∈ Rn : x n = 0}. We may suppose that the ball B(a, r ) lies below 50 . In this case 3 also lies below 50 . Since I is a conformal map we have I ∗ g = v 4/(n−2) δ, where v is the Kelvin transform of u on Rn \ (B(a, r ) ∪ 3). Thus this metric has constant positive scalar curvature n(n−1) in Rn \(B(a, r )∪ 3) and nonnegative mean curvature h on 6. As before, v is a solution of the problem n(n −2) (n+2)/(n−2) =0 in Rn \ (B(a, r ) ∪ 3), v 4 ∂v n −2 n −2 (n)/(n−2) + v+ hv = 0 on 6. ∂ν 2r 2 Also, by hypotheses of contradiction, the mean curvature of the hyperplane 50 at I (q) (with respect to ∂/∂ x n ) is H ≤ 0. By applying the boundary equation of the system (1) to 50 , we obtain 1v +

∂v n −2 + H v n/(n−2) = 0 ∂xn 2 on 50 . Thus we conclude that ∂v/∂ x n (I (q)) ≥ 0. Now we start with the moving planes method. Given λ ≥ 0 we will denote by xλ the reflection of x with respect to the hyperplane 5λ := {x ∈ Rn : x n = λ} and set λ = {x ∈ Rn \ (B(a, r ) ∪ 3) : x n ≤ λ}. We define wλ (x) = v(x) − vλ (x) for x ∈ λ , where vλ (x) := v(xλ ). Since infinity is a regular point of I ∗ g, we have P v(x) = |x|2−n a + bi x i |x|−2 + O(|x|−n ) in a neighborhood of infinity. It follows from [Caffarelli et al. 1989, Lemma 2.3] that there exist R > 0 and λ > 0 such that wλ > 0 in interior of λ \ B(0, R) if λ ≥ λ. Without loss of generality, we can choose R > 0 such that B(a, r ) ∪ 3 ⊂ B(0, R). Now we note that v has a positive infimum, say v0 > 0, in B(0, R)\(B(a, r )∪3). It follows from the fact that v is a classical solution to (5) in B(0, R)\(B(a, r )∪3). So, since v decays in a neighborhood of infinity, we may choose λ > 0 large enough so that vλ (x) < v0 /2 for x ∈ B(0, R) and for λ ≥ λ. Thus for sufficiently large λ, we get wλ > 0 in int(λ ). We also write (7)

1wλ + cλ (x)wλ = 0

in int(λ ),

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where

n(n − 2) v(x)(n+2)/(n−2) − vλ (x)(n+2)/(n−2) . 4 v(x) − vλ (x) By definition, wλ always vanishes on 5λ . In particular, setting λ0 = inf{λ > 0 : wλ > 0 on int(λ ) for all λ ≥ λ} we obtain by continuity that wλ0 satisfies (7), wλ0 ≥ 0 in λ0 , and wλ0 = 0 on 5λ0 . Hence, by applying the strong maximum principle, we conclude that either wλ0 > 0 in int(λ0 ) or wλ0 = v − vλ0 vanishes identically. We point out that the second case occurs only if 3 = ∅. If wλ0 ≡ 0, then 5λ0 is a hyperplane of symmetry of v and therefore v extends to a global positive solution of (5) on the entire Rn . Using [Caffarelli et al. 1989], we conclude that (B1 , g) is a convex spherical cap and the result is obvious. If wλ0 > 0 in int(λ0 ) we apply the E. Hopf maximum principle to conclude cλ (x) =

∂v ∂wλ0 = 2 < 0 in 5λ0 , ∂xn ∂xn and since ∂v/∂ x n (I (q)) ≥ 0, we have λ0 > 0. In this case, by definition of λ0 , we can choose sequences λk ↑ λ0 and xk ∈ λk such that wλk (xk ) < 0. It follows from the work in [Korevaar et al. 1999] that wλ achieves its infimum. Then we lose no generality in assuming xk is a minimum of wλk in λk . We have xk ∈ / 5k because wλk always vanishes on 5λk . So, either xk is in 6 or it is an interior point. Even when xk is an interior point we claim that the xk form a bounded sequence. More precisely: (8)

Claim 3.1 [Chen and Lin 1998, Section 2]. There exists R0 > 0, independent of λ, such that if wλ solves (7) and is negative somewhere in int() and if x0 ∈ int() is a minimum point of wλ , then |x0 | < R0 . For completeness we present a proof in the Appendix. So, we can take a convergent subsequence xk → x ∈ λ0 . Since wλk (xk ) < 0 and wλ0 ≥ 0 in λ0 , we necessarily have wλ0 (x) = 0 and therefore x ∈ ∂λ0 = 5λ0 ∪6. If x ∈ 5λ0 then xk is an interior minimum point to wλk , and hence ∇wλ0 (x) = 0, which cannot occur by inequality (8). Thus we have x ∈ 6 and, by the E. Hopf maximum principle again, (9)

∂wλ0 ∂v ∂v (x) = (x) − (x λ0 ) < 0, ∂η ∂η ∂η

where η := − ν is the inward unit normal vector to 6. Now, we recall that ∂v n −2 n −2 (n+2)/(n−2) + v+ hv = 0 on 6. ∂ν 2r 2 Thus, since v(x) = v(x λ0 ) we have from (9) and (10) that the mean curvature of 6λ0 at x λ0 (with respect to the inward unit normal vector) is strictly less than −h. (10)


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Since h is nonnegative, x λ0 is a nonconvex point in the reflected sphere 6λ0 Considering the problem back to B1 , we denote by K 1 the ball corresponding to the one whose boundary is 6λ0 and by P1 the ball corresponding to 5+ λ0 . Thus we have obtained a strictly smaller ball K 1 ⊂ B with a nonconvex boundary which is the reflection of ∂ B1 with respect to ∂ P1 . We can repeat this argument to obtain a sequence of balls with nonconvex points on the boundaries, that is, B ⊃ K 1 ⊃ · · · ⊃ K j ⊃ · · · . This sequence cannot converge to a point, since small balls are always convex. On the other hand, if K j → K ∞ where K ∞ is not a point, then K ∞ ⊂ B is a ball in B1 \ 3 whose boundary is the reflection of ∂ B1 with respect to itself. This is a contradiction. Appendix. Proof of Claim 3.1 First write Equation (7), setting cλ (x) = 0 when wλ (x) = 0. Fix 0 < µ < n − 2, and define g(x) = |x|−µ and φ(x) = wλ (x)/g(x). Then, using (7), 1g 2 φ = 0. 1φ + h∇g, ∇φi + cλ (x) + g g By a computation we get 1g = −µ(n − 2 − µ)|x|−µ−2 , that is, 1g = −µ(n − 2 − µ)|x|−2 . g On the other hand, the expansion of v in a neighborhood of infinity implies that wλ (x) = O(|x|2−n ) and consequently cλ (x) = O(|x|−n−2−2+n ) = O(|x|−4 ). Hence we obtain 1g cλ (x) + ≤ C(|x|−4 − µ(n − 2 − µ))|x|−2 ). g In particular c(x) + 1g/g < 0 for large |x|. Choose R0 with B(a, r ) ∪ 3 ⊂ B(0, R0 ) such that (11)

C(|x|−4 − µ(n − 2 − µ))|x|−2 ) < 0

for |x| ≥ R0 .

Now let x0 ∈ int(λ ) so that wλ (x0 ) = infint(λ ) wλ < 0. Since lim|x|→+∞ φ(x) = 0 and φ(x) ≥ 0 on ∂λ , there exists x 0 such that φ has its minimum at x 0 . By applying the maximum principle for φ at x 0 we get cλ (x 0 ) + 1g(x 0 )/g ≥ 0 and by (11), we get |x 0 | < R0 . Now we have wλ (x0 ) wλ (x 0 ) wλ (x0 ) ≤ = φ(x 0 ) ≤ φ(x0 ) = . g(x 0 ) g(x 0 ) g(x0 ) This implies |x0 | ≤ |x 0 | ≤ R0 and proves the claim.

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Acknowledgements The content of this paper is part of the author’s doctoral thesis [Cavalcante 2006]. The author would like to express his gratitude to Professor Manfredo do Carmo for the encouragement and to Professor Fernando Coda Marques for many useful discussions during this work. While the author was at IMPA in Rio de Janeiro, he was fully supported by CNPq-Brazil. References [Aubin 1976] T. Aubin, “Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire”, J. Math. Pures Appl. (9) 55:3 (1976), 269–296. MR 55 #4288 Zbl 0336.53033 [Aviles and McOwen 1988] P. Aviles and R. C. McOwen, “Complete conformal metrics with negative scalar curvature in compact Riemannian manifolds”, Duke Math. J. 56:2 (1988), 395–398. MR 89b:58224 Zbl 0645.53023 [Caffarelli et al. 1989] L. A. Caffarelli, B. Gidas, and J. Spruck, “Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth”, Comm. Pure Appl. Math. 42:3 (1989), 271–297. MR 90c:35075 Zbl 0702.35085 [Cavalcante 2006] M. P. A. Cavalcante, Conformally flat metrics, constant mean curvature surfaces in product spaces and the r -stability of hypersurfaces., PhD thesis, Instituto Nacional de Matemática Pura e Aplicada, 2006. [Chen and Lin 1998] C.-C. Chen and C.-S. Lin, “Estimate of the conformal scalar curvature equation via the method of moving planes. II”, J. Differential Geom. 49:1 (1998), 115–178. MR 2000h:35045 Zbl 0961.35047 [Escobar 1992a] J. F. Escobar, “Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary”, Ann. of Math. (2) 136:1 (1992), 1–50. MR 93e:53046 Zbl 0766.53033 [Escobar 1992b] J. F. Escobar, “The Yamabe problem on manifolds with boundary”, J. Differential Geom. 35:1 (1992), 21–84. MR 93b:53030 Zbl 0771.53017 [Finn and McOwen 1993] D. L. Finn and R. C. McOwen, “Singularities and asymptotics for the equation 1g u − u q = Su”, Indiana Univ. Math. J. 42:4 (1993), 1487–1523. MR 95g:58216 Zbl 0791.35010 [Gidas et al. 1979] B. Gidas, W. M. Ni, and L. Nirenberg, “Symmetry and related properties via the maximum principle”, Comm. Math. Phys. 68:3 (1979), 209–243. MR 80h:35043 Zbl 0425.35020 [Han and Li 1999] Z.-C. Han and Y. Li, “The Yamabe problem on manifolds with boundary: existence and compactness results”, Duke Math. J. 99:3 (1999), 489–542. MR 2000j:53045 Zbl 0945. 53023 [Korevaar et al. 1999] N. Korevaar, R. Mazzeo, F. Pacard, and R. Schoen, “Refined asymptotics for constant scalar curvature metrics with isolated singularities”, Invent. Math. 135:2 (1999), 233–272. MR 2001a:35055 Zbl 0958.53032 [Marques 2005] F. C. Marques, “Existence results for the Yamabe problem on manifolds with boundary”, Indiana Univ. Math. J. 54:6 (2005), 1599–1620. MR 2006j:53047 Zbl 05014742 [McOwen 1993] R. C. McOwen, “Singularities and the conformal scalar curvature equation”, pp. 221–233 in Geometric analysis and nonlinear partial differential equations (Denton, TX, 1990), Lect. Notes in Pure and Appl. Math. 144, Dekker, New York, 1993. MR 94b:53076 Zbl 0826.58034


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[Pollack 1993] D. Pollack, “Compactness results for complete metrics of constant positive scalar curvature on subdomains of S n ”, Indiana Univ. Math. J. 42:4 (1993), 1441–1456. MR 95c:53052 Zbl 0794.53025 [Schoen 1984] R. Schoen, “Conformal deformation of a Riemannian metric to constant scalar curvature”, J. Differential Geom. 20:2 (1984), 479–495. MR 86i:58137 Zbl 0576.53028 [Schoen 1991] R. M. Schoen, “On the number of constant scalar curvature metrics in a conformal class”, pp. 311–320 in Differential geometry, Pitman Monogr. Surveys Pure Appl. Math. 52, Longman Sci. Tech., Harlow, 1991. MR 94e:53035 Zbl 0733.53021 [Trudinger 1968] N. S. Trudinger, “Remarks concerning the conformal deformation of Riemannian structures on compact manifolds”, Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 265–274. MR 39 #2093 Zbl 0159.23801 [Yamabe 1960] H. Yamabe, “On a deformation of Riemannian structures on compact manifolds”, Osaka Math. J. 12 (1960), 21–37. MR 23 #A2847 Zbl 0096.37201 Received January 19, 2007. Revised June 8, 2007. ´ M ARCOS P ETR UCIO DE A. C AVALCANTE ´ I NSTITUTO DE M ATEM ATICA U NIVERSIDADE F EDERAL DE A LAGOAS ˜ , BR 104 N ORTE , K M 97 C AMPUS A. C. S IM OES 57072-970 M ACEI O´ , AL B RAZIL [email protected]


BRAID GROUP REPRESENTATIONS FROM TWISTED QUANTUM DOUBLES OF FINITE GROUPS PAVEL E TINGOF , E RIC ROWELL

AND

S ARAH W ITHERSPOON

We investigate the braid group representations arising from categories of representations of twisted quantum doubles of finite groups. For these categories, we show that the resulting braid group representations always factor through finite groups, in contrast to the categories associated with quantum groups at roots of unity. We also show that in the case of p-groups, the corresponding pure braid group representations factor through a finite pgroup, which answers a question asked of the first author by V. Drinfeld.

1. Introduction Any braided tensor category C gives rise to finite dimensional representations of the braid group Bn . A natural problem is to determine the image of these representations. This has been carried out to some extent for the braided tensor categories coming from quantum groups and polynomial link invariants at roots of unity [Birman and Wajnryb 1986; Franko et al. 2006; Freedman et al. 2002; Goldschmidt and Jones 1989; Jones 1986; 1989; Larsen and Rowell ≥ 2008; Larsen et al. 2005]. A basic question in this direction is this: Is the image of the representation of Bn a finite group? In the aforementioned papers the answer is typically “no”, as finite groups appear only in a few cases when the degree of the root of unity is small. Here, we consider the braid group representations associated to the (braided tensor) categories Mod-D ω (G), where D ω (G) is the twisted quantum double of the finite group G. We show in Theorem 4.2 that the braid group images are always finite. In Theorem 4.5, we also answer in the affirmative a question of Drinfeld: If G is a p-group, is the image of the pure braid group Pn also a p-group? The content of the paper is as follows. In Section 2 we record some definitions and basic results on braided categories, and Section 3 is dedicated to the needed facts about D ω (G). Then we prove our main results in Section 4. The last section describes some open problems suggested by our work. MSC2000: primary 16W30; secondary 20F36, 18D10. Keywords: twisted quantum doubles, Artin’s braid group, pure braid group, representations, modular categories. 33

34

PAVEL ETINGOF, ERIC ROWELL

AND

SARAH WITHERSPOON

2. Braided categories and braid groups We recall some facts about braided categories and derive some basic consequences. For more complete definitions the reader is referred to either [Bakalov and Kirillov 2001] or [Turaev 1994]. The braid group Bn is defined by generators β1 , . . . βn−1 satisfying the relations (B1)

βi βi+1 βi = βi+1 βi βi+1 for 1 ≤ i ≤ n − 2, and

(B2)

βi β j = β j βi if |i − j| ≥ 2.

The kernel of the surjective homomorphism from Bn to the symmetric group Sn given by βi 7→ (i, i + 1) is the pure braid group Pn , and it is generated by all conjugates of β12 . Let C be a k-linear braided category over an algebraically closed field k of arbitrary characteristic. The braiding structure affords us representations of Bn as follows. For any object X in C we have braiding isomorphisms c X,X ∈ End(X ⊗2 ), so that defining ∈ End(X ⊗n ), ⊗c X,X ⊗ Id⊗(n−i−1) Rˇ i := Id⊗(i−1) X X we obtain a representation φ nX of Bn by automorphisms of X ⊗n by φ nX (βi ) = Rˇ i . n Similarly, for any collection of objects {X i }i=1 , one has representations of Pn on X 1 ⊗ · · · ⊗ X n . Throughout the paper, when we refer to representations of Pn and Bn arising from tensor products of objects in a braided category, these are the representations we mean. We say that Y is a subobject of Z if there is a monomorphism q ∈ HomC (Y, Z ) and W is a quotient object of Z if there is an epimorphism p ∈ HomC (Z , W ). Because of the functoriality of the braiding, we have the following obvious lemma, which will be used in Section 4.

Lemma 2.1. (i) If Y is a quotient object or a subobject of Z , then φYn (Bn ) is a quotient group of φ nZ (Bn ) and a similar statement holds for the restrictions of these representations to Pn . (ii) Let S be a finite set of objects of a braided tensor category C for which the image of the representation of Pn in End(X 1 ⊗ · · · ⊗ X n ) is finite for all X 1 , . . . , X n ∈ S. Let X be the direct sum of finitely many objects taken from S. Then the image of the representation of Bn in End(X ⊗n ) is finite.

BRAID GROUP REPRESENTATIONS FROM TWISTED QUANTUM DOUBLES

35

3. The twisted quantum double of a finite group In this section we define the twisted quantum double of a finite group and give some basic results that we need. For more details, see for example [Chari and Pressley 1994; Drinfeld 1989; Witherspoon 1996]. Let k be an algebraically closed field of arbitrary characteristic `. Let G be a finite group with identity element e, let kG be the corresponding group algebra, and let (kG)∗ be the dual algebra of linear functions from kG to k, under pointwise multiplication. There is a basis of (kG)∗ consisting of the dual functions δg for g ∈ G defined by δg (h) = δg,h (g, h ∈ G). Let ω : G × G × G → k × be a 3-cocycle, that is, ω(a, b, c)ω(a, bc, d)ω(b, c, d) = ω(ab, c, d)ω(a, b, cd) for all a, b, c, d ∈ G. The twisted quantum (or Drinfeld) double D ω (G) is a quasi-Hopf algebra whose underlying vector space is (kG)∗ ⊗ kG. We abbreviate the basis element δx ⊗ g of D ω (G) by δx g for x, g ∈ G. Multiplication on D ω (G) is defined by (δx g)(δ y h) = θx (g, h)δx,gyg−1 δx gh, where θx (g, h) =

ω(x, g, h)ω(g, h, h −1 g −1 xgh) . ω(g, g −1 xg, h)

As an algebra, D ω (G) is semisimple if and only if the characteristic ` of k does not divide the order of G [Witherspoon 1996]. The quasi-coassociative coproduct 1 : D ω (G) → D ω (G) ⊗ D ω (G) is defined by X 1(δx g) = γg (y, z)δ y g ⊗ δz g, y,z∈G yz=x

where γg (y, z) =

ω(y, z, g)ω(g, g −1 yg, g −1 zg) . ω(y, g, g −1 zg)

The quasi-Hopf algebra D ω (G) is quasitriangular with X X R= δg ⊗ g and R −1 = θghg−1 (g, g −1 )−1 δg e ⊗ δh g −1 . g∈G

g,h∈G

In particular R1(a)R −1 = σ (1(a)) for all a ∈ D ω (G), where σ is the transposition map. If X and Y are D ω (G)-modules, then Rˇ = σ ◦ R provides a D ω (G)-module ˇ Then the isomorphism from X ⊗ Y to Y ⊗ X . Let c X,Y be this action by R. ω ω category Mod-D (G) of finite dimensional D (G)-modules is a braided category with braiding c.

36


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SARAH WITHERSPOON

4. The images of Bn and Pn In this section we fix a finite group G and a 3-cocycle ω, and we prove that the image of Bn in End D ω (G) (V ⊗n ) is finite for any positive integer n and any finite dimensional D ω (G)-module V . When G is a p-group, we prove that the image of Pn in End D ω (G) (V ⊗n ) is also a p-group. Remark 4.1. It follows from a theorem of C. Vafa, see [Bakalov and Kirillov 2001, Theorem 3.1.19], and the so-called balancing axioms that for braided fusion categories over C, the images of the braid group generators βi in the above representations of Bn always have finite order. This is far from enough to conclude that the image of Bn is finite; Coxeter [1959] has shown that the quotient of Bn by the normal closure of the subgroup generated by {βik : 1 ≤ i ≤ n − 1} is finite if and only if 1/n + 1/k > 1/2. The case of general finite groups. Let r and m be positive integers. The full monomial group G(r, 1, m) is the multiplicative group consisting of the m × m matrices having exactly one nonzero entry in each row and column, all of whose nonzero entries are r -th roots of unity. It is one of the irreducible complex reflection groups. Let r = |G|`0 be the part of |G| not divisible by the characteristic ` of k, that is, |G| = r `s and (r, `) = 1. Theorem 4.2. Let V be a finite dimensional D ω (G)-module. Then the image of Bn in End(V ⊗n ) is finite. More specifically, this image is a quotient of a subgroup of G(r, 1, m) for m = |G|2n . Proof. We will need the following well-known lemma, which follows from [Weibel 1994, Theorem 6.5.8]. Let µr ⊂ k × be the set of r -th roots of unity. Lemma 4.3. The natural map H i (G, µr ) → H i (G, k × ) is surjective. In particular, any element in H i (G, k × ) may be represented by a cocycle taking values in µr . Now we turn to the proof of the theorem. As any finite dimensional D ω (G)module is finitely generated and is therefore a quotient of a finite rank free module, by Lemma 2.1(i) it suffices to prove the statement when V is a finite-rank free module. By Lemma 2.1(ii), we need only consider the case V = D ω (G), the left regular module. Assume first that n = 2. Let x, y, a, b ∈ G. The action of Rˇ on the basis element δx a ⊗ δ y b of D ω (G) ⊗ D ω (G) is X ˇ x a ⊗ δ y b) = σ R(δ δg ⊗ g (δx a ⊗ δ y b) = σ (θx yx −1 (x, b)δx a ⊗ δx yx −1 xb) g∈G

= θx yx −1 (x, b)δx yx −1 xb ⊗ δx a.


37

If n > 2, similar calculations show that each Rˇi permutes the chosen basis of up to scalar multiples of the form θx yx −1 (x, b). By Lemma 4.3, we may assume that ω and hence θ takes values in the r -th roots of unity. This implies that the image of Bn in End(D ω (G)⊗n ) is contained in G(r, 1, m). D ω (G)

Corollary 4.4. Let C be a braided fusion category that is group theoretical in the sense of [Etingof et al. 2005]. Let V be any object of C. Then the image of Bn in End(V ⊗n ) is finite. Proof. Let Z (C) be the Drinfeld center of C. Since C is braided, we have a canonical braided tensor functor F : C → Z (C). Thus it suffices to show the result holds for the category Z (C). Since C is group theoretical, Z (C) is equivalent to Mod -D ω (G) for some G, ω; see [Natale 2003]. Thus the desired result follows from Theorem 4.2. The case of p-groups. Theorem 4.5. Suppose that G is a finite p-group and V is a finite dimensional D ω (G)-module. Then the image of Pn in End(V ⊗n ) is also a p-group. The rest of the subsection is occupied by the proof of Theorem 4.5. We will need a technical lemma: Lemma 4.6. Let H be a group with normal subgroups H = H0 ⊃ H1 ⊃ ... ⊃ HN = 1 such that Hi /Hi+1 is abelian and [Hi , H j ] ⊂ Hi+ j , and let I be a subgroup of Aut(H ) that preserves this filtration and acts trivially on the associated graded group. Then I is nilpotent of class at most N − 1. Proof. Let L 1 (I ) = I , L 2 (I ) = [I, I ], L 3 (I ) = [[I, I ], I ], . . . be the lower central series of I . We must show L N (I ) = 1. We prove by induction on n that for any f ∈ L n (I ) and h ∈ Hm , we have f (h) = ha(h), where a(h) ∈ Hn+m . The case n = 1 is clear, since f ∈ I acts trivially on Hm /Hm+1 . Suppose the statement is true for n. Take g ∈ I , f ∈ L n (I ), and h ∈ Hm such that f (h) = ha(h) and g(h) = hb(h), where a(h) ∈ Hn+m and b(h) ∈ Hm+1 . Then f g(h) = f (h) f (b(h)) = ha(h)b(h)a(b(h)), while g f (h) = hb(h)a(h)b(a(h)). Since g acts trivially on the associated graded group, we have b(a(h)) ∈ Hn+m+1 . Also a(b(h)) ∈ Hn+m+1 since b(h) ∈ Hm+1 , by the induction assumption. Also a(h)b(h) = b(h)a(h) modulo Hn+m+1 since [Hi , H j ] ⊂ Hi+ j . Thus f g(h) = g f (h) in H/Hn+m+1 , and thus [ f, g](h) = h in H/Hn+m+1 , which is what we needed to show. Taking m = 0 and n = N − 1, any [ f, g] ∈ L N (I ) is the identity on H = H/HN , and the lemma is proved.

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Proof of Theorem 4.5. Any finite dimensional D ω (G)-module is a quotient of a multiple of the left regular D ω (G)-module H = D ω (G). By Lemma 2.1, it suffices to show that the image of Pn in End(H ⊗n ) is a p-group. By Theorem 4.2, the image K of Pn is a subgroup of the full monomial group G(r, 1, m), where r = p t for some t, and m = |G|2n . The normal subgroup of diagonal matrices in K is thus a p-group, and so it is enough to show that K modulo the diagonal matrices is a p-group. Thus it suffices to assume that ω = 1 and H = D(G). Computing, we have X ˇ x ⊗ bδ y ) = σ R(aδ δg ⊗ g (aδx ⊗ bδ y ) = axa −1 b δ y ⊗ aδx g∈G

for all a, b, x, y ∈ G. Denote by (g, x) the element gδx so that a basis of H ⊗n is (g1 , x1 ) ⊗ · · · ⊗ (gn , xn ) with gi , xi ∈ G. The braid generator βi fixes all factors other than the i-th and (i+1)-st, and on these it acts by (gi , xi ) ⊗ (gi+1 , xi+1 ) 7→ (gi xi gi−1 xi+1 , xi+1 ) ⊗ (gi , xi ) = ([gi , xi ]xi gi+1 , xi+1 ) ⊗ (gi , xi ), where [a, b] denotes the group commutator. This action induces a homomorphism ψ : Bn → Aut(Fr2n ) where Fr2n is the free group on 2n generators. Explicitly, n ψ(βi ) is the automorphism defined on generators {gi , xi }i=1 of Fr2n by x j 7→ x j , xi 7→ xi+1 ,

g j 7→ g j

for j 6∈ {i, i + 1},

xi+1 7→ xi ,

gi 7→ [gi , xi ]xi gi+1 ,

gi+1 7→ gi .

Since G is a p-group, it is nilpotent of class, say, N − 1. Note that ψ descends to a homomorphism ψ N : Bn → Aut(Fr2n /L N (Fr2n )) where L N (Fr2n ) denotes the N -th term of the lower central series of Fr2n . Since G is nilpotent of class N − 1, the action of Bn on the set G 2n defined above factors through ψ N . Thus, setting I = ψ N (Pn ), one sees that the action of Pn on H ⊗n factors through I , that is, K is a quotient of I . Let us now show that I is nilpotent. Define a descending filtration on M = Fr2n /L N (Fr2n ) by positive integers as follows. Let M1 = M. Define degrees on the generators by deg(gi ) = 1 and deg(xi ) = 2 for all i, and define M j for j ≥ 2 to be the normal closure of the group generated by [Mk , M j−k ] for all 0 ≤ k ≤ j together with the generators of degree at least j. Since M is nilpotent, this filtration is finite. Further, I preserves this filtration and acts trivially on the quotients Mi /Mi+1 . By Lemma 4.6, I is nilpotent.


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It follows that the finite group K is nilpotent. However, K is generated by conjugates of β12 , and we claim that β12 is an element whose order is a power of p. Indeed, this follows from the fact that if the ground field is C (which may be assumed without loss of generality, since the double of G is defined over the integers), then the eigenvalues of c X,Y cY,X for any objects X and Y are ratios of twists, which are computed from characters of G (in [Bakalov and Kirillov 2001]), and hence are roots of unity whose degrees are powers of p. Therefore, K is a finite p-group. 5. Questions We mention some directions for further investigation suggested by these (and other) results. We refer the reader to [Etingof et al. 2005] and [Turaev 1994] for the relevant definitions. (1) Suppose G is a p-group. Theorem 4.5 shows that the image of the associated representation of Pn is also a p-group. What is its nilpotency class relative to that of G? Some upper bounds can be obtained from the proof of Theorem 4.5, but it is not clear how tight they are. (2) The finite groups that appear as images of representations of Bn associated to quantum groups and link invariants at roots of unity (see [Franko et al. 2006; Goldschmidt and Jones 1989; Jones 1986; 1989; Larsen and Rowell ≥ 2008; Larsen et al. 2005]) basically fall into two classes: symplectic groups and extensions of p-groups by the symmetric group Sn . Does this hold for the representations of Bn associated with Mod -D ω (G)? In general, is there a relationship between the image of Pn and G? (3) As a modular category, Mod -D ω (G) gives rise to (projective) representations of mapping class groups of compact surfaces with boundary. Are the images always finite? It is known to be true for the mapping class groups of the torus and the n-punctured sphere (Theorem 4.2). For more general modular categories, the answer is definitely “no.” See [Andersen et al. 2006, Conjecture 2.4]. (4) Let us say that a braided category C has property F if all braid group representations associated to C have finite images. What class of braided categories have property F? Among braided fusion categories, Corollary 4.4 shows that all braided group-theoretical categories, in the sense of [Etingof et al. 2005], have property F. Do all braided fusion categories with integer Frobenius– Perron dimension have property F?

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Acknowledgments Etingof is grateful to V. Drinfeld for a useful discussion and for raising the question answered by Theorem 4.5. His work was partially supported by NSF grant DMS0504847. Witherspoon was partially supported by NSA grant H98230-07-1-0038. References [Andersen et al. 2006] J. E. Andersen, G. Masbaum, and K. Ueno, “Topological quantum field theory and the Nielsen–Thurston classification of M(0, 4)”, Math. Proc. Cambridge Philos. Soc. 141:3 (2006), 477–488. MR 2007k:57059 Zbl 1110.57009 [Bakalov and Kirillov 2001] B. Bakalov and A. Kirillov, Jr., Lectures on tensor categories and modular functors, University Lecture Series 21, American Mathematical Society, Providence, RI, 2001. MR 2002d:18003 Zbl 0965.18002 [Birman and Wajnryb 1986] J. S. Birman and B. Wajnryb, “Markov classes in certain finite quotients of Artin’s braid group”, Israel J. Math. 56:2 (1986), 160–178. MR 88b:20063 Zbl 0621.20025 [Chari and Pressley 1994] V. Chari and A. Pressley, A guide to quantum groups, Cambridge University Press, Cambridge, 1994. MR 95j:17010 Zbl 0839.17009 [Coxeter 1959] H. S. M. Coxeter, “Factor groups of the braid group”, pp. 95–122 in Proceedings of the Fourth Can. Math. Cong. (Banff 1957), University of Toronto Press, 1959. Zbl 0093.25003 [Drinfeld 1989] V. G. Drinfeld, “Quasi-Hopf algebras”, Algebra i Analiz 1:6 (1989), 114–148. MR 91b:17016 [Etingof et al. 2005] P. Etingof, D. Nikshych, and V. Ostrik, “On fusion categories”, Ann. of Math. (2) 162:2 (2005), 581–642. MR 2006m:16051 Zbl 05042683 [Franko et al. 2006] J. M. Franko, E. C. Rowell, and Z. Wang, “Extraspecial 2-groups and images of braid group representations”, J. Knot Theory Ramifications 15 (2006), 413–427. MR 2006m:20053 Zbl 1097.20034 [Freedman et al. 2002] M. H. Freedman, M. J. Larsen, and Z. Wang, “The two-eigenvalue problem and density of Jones representation of braid groups”, Comm. Math. Phys. 228:1 (2002), 177–199. MR 2004d:20037 Zbl 1045.20027 [Goldschmidt and Jones 1989] D. M. Goldschmidt and V. F. R. Jones, “Metaplectic link invariants”, Geom. Dedicata 31:2 (1989), 165–191. MR 91e:57014 Zbl 0678.57007 [Jones 1986] V. F. R. Jones, “Braid groups, Hecke algebras and type II1 factors”, pp. 242–273 in Geometric methods in operator algebras (Kyoto, 1983), edited by H. Araki and E. G. Effros, Pitman Res. Notes Math. Ser. 123, Longman Sci. Tech., Harlow, 1986. MR 88k:46069 Zbl 0659.46054 [Jones 1989] V. F. R. Jones, “On a certain value of the Kauffman polynomial”, Comm. Math. Phys. 125:3 (1989), 459–467. MR 91e:57017 Zbl 0695.57003 [Larsen and Rowell ≥ 2008] M. J. Larsen and E. C. Rowell, “An algebra-level version of a linkpolynomial identity of Lickorish”, Math. Proc. Cambridge Philos. Soc.. To appear. [Larsen et al. 2005] M. J. Larsen, E. C. Rowell, and Z. Wang, “The N -eigenvalue problem and two applications”, Int. Math. Res. Not. 64 (2005), 3987–4018. MR 2006m:22019 Zbl 05030168 [Natale 2003] S. Natale, “On group theoretical Hopf algebras and exact factorizations of finite groups”, J. Algebra 270:1 (2003), 199–211. MR 2004k:16102 Zbl 1040.16027 [Turaev 1994] V. G. Turaev, Quantum invariants of knots and 3-manifolds, de Gruyter Studies in Mathematics 18, Walter de Gruyter & Co., Berlin, 1994. MR 95k:57014 Zbl 0812.57003


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[Weibel 1994] C. A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge University Press, 1994. MR 95f:18001 Zbl 0797.18001 [Witherspoon 1996] S. J. Witherspoon, “The representation ring of the twisted quantum double of a finite group”, Canad. J. Math. 48:6 (1996), 1324–1338. MR 97m:16078 Zbl 0876.16027 Received March 9, 2007. PAVEL E TINGOF D EPARTMENT OF M ATHEMATICS M ASSACHUSETTS I NSTITUTE OF T ECHNOLOGY 77 M ASSACHUSETTS AVE . C AMBRIDGE , MA 02139-4307 [email protected] E RIC ROWELL D EPARTMENT OF M ATHEMATICS T EXAS A&M U NIVERSITY C OLLEGE S TATION , TX 77843-3368 [email protected] S ARAH W ITHERSPOON D EPARTMENT OF M ATHEMATICS T EXAS A&M U NIVERSITY C OLLEGE S TATION , TX 77843-3368 [email protected]


INTERIOR AND BOUNDARY REGULARITY OF INTRINSIC BIHARMONIC MAPS TO SPHERES Y IN B ON K U The interior and boundary regularity of weakly intrinsic biharmonic maps from 4-manifolds to spheres is proved.

1. Introduction The regularity problem of harmonic maps has been intensively studied for a long time. Hélein [1991] proved that any weakly harmonic maps from a closed Riemannian surface to a compact Riemannian manifold without boundary is smooth. Later Qing [1993] proved the boundary regularity for weakly harmonic maps from compact Riemannian surface with boundary. However, when the domain dimension is greater than 2, Rivière [1995] constructed everywhere discontinuous weakly harmonic maps into spheres. This implies that there is no hope of getting any regularity results for weakly harmonic maps in higher dimensional cases. Therefore, it is of interest to study higher order energy functionals that enjoy better regularity properties. Let M be a Riemmanian manifold and N be a compact Riemannian manifold without boundary that is isometrically embedded in R K . We say that u is biharmonic map if it is a critical point of the functional F(v) = Ra weaklyTintrinsic 2 2,2 (M, N ), where (1v)T is the component of 1v in R K M |(1v) | for v ∈ W that is tangent to N at v( p) ∈ N for all p ∈ M. (Sometimes it is called the tension field τ (v) in the literature.) If the critical point u is smooth, we say u is an intrinsic biharmonic map. It is intrinsic in that the definition is independent of the choice of isometric embedding of the N into R K . If u ∈ W 2,2 (M, N ) is a weakly harmonic map, then (1u)T = 0, and therefore u is obviously a minimizer of F. In other words, the class of all weakly intrinsic biharmonic maps can be regarded as an extension of the class of all weakly harmonic maps in W 2,2 (M, N ).R Another functional considered by Chang, Wang, and Yang [1999c] is FE (v) = M |1v|2 , whose critical point is called a weakly extrinsic biharmonic map. Unlike a intrinsic biharmonic map, it depends on the choice of the embedding. MSC2000: primary 58E20; secondary 35G20. Keywords: biharmonic maps, regularity, higher-order PDE. 43

44

YIN BON KU

The interior regularity of weakly intrinsic and extrinsic biharmonic maps from a bounded domain in R4 to a compact Riemannian manifold without boundary was established by C. Wang [2004]. And in recent paper of Lamm and Riviére [≥ 2008], they successfully rewrite the Euler–Lagrange equation of a weakly intrinsic and extrinsic biharmonic map into a conservation law, which simplifies the proof of interior regularity. However, it remains unclear whether this method can be used to prove the boundary regularity. Here, we use the idea from [Chang et al. 1999c] to prove the interior and boundary regularity of weakly intrinsic biharmonic maps from four-dimensional Riemannian manifolds to S n in Rn+1 , that is, if u ∈ W 2,2 (M, S n ) is weakly intrinsic biharmonic, then it is intrinsic biharmonic. Moreover, if u has smooth Dirichlet boundary data on ∂ M, then it is smooth up to the boundary. The paper is arranged as follows. In Section 2, we introduce necessary notations and derive the explicit Euler–Lagrange equations of a weakly intrinsic biharmonic map to S k ; the equations make up a fourth-order nonlinear elliptic system. As in [Chang et al. 1999c], by exploiting the special structure of the nonlinearity of these Euler–Lagrange equations, we are able to rewrite them as 12 u = a linear combination of several special types of “divergence forms.” From this, we can obtain the crucial L p estimate which is key to the proof of interior Hölder regularity of u. In Section 3, we prove that if u is Hölder continuous, it must be smooth. The proof is based on an interesting observation in [Chang et al. 1999b] that if u is continuous, the coefficients of the nonlinear terms can be made very small by a specific scaling. Then by an iteration process, we prove that second derivatives of u are Hölder continuous. Now standard regularity theory implies that u is smooth, hence completing the proof of the interior regularity theorem. In Section 4, we prove the boundary regularity theorem by modifying the method of proof of interior regularity. For simplicity, we assume throughout the paper that the domain of the intrinsic biharmonic map is a flat Euclidean ball. The proof in the general case is essentially the same. The author thanks Professor Alice Chang, Professor Paul Yang, and Professor Lihe Wang for their helpful suggestions.

2. Interior Hölder regularity Here, we consider the interior Hölder regularity of a weakly intrinsic biharmonic map u. Since this is a local property, we may assume without loss of generality that u : (B, g) → S n ⊂ Rn+1 , where B is a 4-dimensional unit ball in R4 with Euclidean metric and S n is canonically embedded in Rn+1 with the induced standard metric. 1, ∇, and div denote the Laplacian, gradient, and divergence.

REGULARITY OF INTRINSIC BIHARMONIC MAPS TO SPHERES

45

2.1. The functional F. Let u ∈ W 2,2 (B, S n ) be a weakly intrinsic biharmonic map. Write u(x) = (u 1 (x), . . . , u n+1 (x)) ∈ Rn+1 for x ∈ B. It is well known that ((1u)T )α = 1u α + u α |∇u|2

for α = 1, 2, . . . , n + 1.

Therefore, by straightforward calculations, we have Z F(u) = (|1u|2 − |∇u|4 ). B

And its Euler–Lagrange equation is P (1u β )2 + 1(|∇u β |2 ) + 2∇1u β · ∇u β + 2|∇u|4 u α (1) 12 u α = − β

− 2 div(|∇u|2 ∇u α ) for α = 1, 2, . . . , n + 1. We say that u ∈ W 2,2 (B, S n ) is weakly intrinsic biharmonic if and only if it satisfies Equation (1) weakly. 2.2. Divergence forms. Now, we are going to write the right hand side of Equation (1) into a linear combination of certain types of “divergence forms.” Using notations in [Chang et al. 1999c], we define T1 ≡ div(∇u α 1u β (u β − cβ )) or div((u β − cβ )h∇∇u β , ∇u β i), T2 ≡ 1((u α − cα )|∇u β |2 ) or 1((u β − cβ )1u β ) or 1(u α (u β − cβ )1u β ), T3 ≡ 1(div((u β − cβ )∇u β )), where cβ are constants and the β are summed from 1 to n + 1. In our case, we have to consider one more type, namely, T4 ≡ div(|∇u|2 (u α ∇u β − u β ∇u α )(u β − cβ )). Proposition 2.1. The right hand side of Equation (1) can be written as a linear combination of Tl terms for l = 1, 2, 3, 4. Proof. At any point p ∈ B, we choose a normal coordinate x = (x1 , . . . , x4 ) at p and let u i be the i-th covariant derivative of u. We name S1 = u α (1u β )2 , β S2 = 2u α u j (1u β ) j , and S3 = u α 1|∇u β |2 . Note that the j are summed from 1 to

46

YIN BON KU

4. Then 1 2 S2

β β = u α u j (1u β ) j = u α (1u β ) j − u β (1u α ) j u j β β = u α (1u β ) j − u β (1u α ) j − u αj (1u β ) + u j (1u α ) u j β β + u αj (1u β ) − u j (1u α ) u j β = u α (1u β ) j − u β (1u α ) j − u αj (1u β ) + u j (1u α ) (u β − cβ ) j β β β α 2 β β 2 α β α β α − u 1 u − u 1 u (u − c ) + u j (1u ) − u j (1u ) u j = u α (1u β ) − u β (1u α ) (u β − cβ ) jj β − 2 u αj (1u β ) − u j (1u α ) (u β − cβ ) j β β β − u α (1u β ) − u β (1u α ) u j j + u αj (1u β ) − u j (1u α ) u j − u α 12 u β − u β 12 u α (u β − cβ ) β β β = − u α (1u β ) − u β (1u α ) u j j + u αj (1u β ) − u j (1u α ) u j − u α 12 u β − u β 12 u α (u β − cβ ) + (T2 + T1 terms).

By [Chang et al. 1999c], we know that S1 + S3 = (u α 1u β − u β 1u α )1u β + (T` terms for ` = 1, 2, 3), β β β = (u α 1u β − u β 1u α )u j j − (u αj 1u β − u j 1u α )u j β − u α (1u β ) j − u β (1u α ) j u j + (T` terms), = − 12 S2 − 21 S2 − u α 12 u β − u β 12 u α (u β − cβ ) + (T` terms), S1 + S2 + S3 = (T` terms) − (u α 12 u β − u β 12 u α )(u β − cβ ) But by (1), we get that u α 12 u β − u β 12 u α = − 2 div(|∇u|2 ∇u β )u α − µu α u β − −2 div(|∇u|2 ∇u α )u β − µu α u β

= − 2 div(|∇u|2 ∇u β )u α + 2 div(|∇u|2 ∇u α )u β β = 2 |∇u|2 (u β u αj − u α u j ) j Hence we have right side of (1) = − (λ + 2|∇u|4 )u α − 2 div(|∇u|2 ∇u α ) β = (T` terms) + 2 |∇u|2 (u β u αj − u α u j ) j (u β − cβ ) − 2|∇u|4 u α


47

β = (T` terms) + 2 |∇u|2 (u β u αj − u α u j ) j (u β − cβ ) β

− 2|∇u|2 (u α u j − u β u αj )(u β − cβ ) j β = (T` terms) − 2 |∇u|2 (u α u j − u β u αj )(u β − cβ ) j = (T` terms for ` = 1, 2, 3, 4). β

The third equality follows from u α |∇u|2 = (u α u j − u β u αj )(u β − cβ ) j .

2.3. Hölder continuity of u. Theorem 2.1. If u ∈ W 2,2 (B, S 4 ) is weakly intrinsic biharmonic, then it is locally Hölder continuous on B with exponent β for some β ∈ (0, 1). To prove this, we first need standard L p elliptic estimates: Lemma 2.1. Suppose Br is a Euclidean ball in R4 of radius r > 0 and v ∈ W 2,2 (Br ) is a weak solution on Br of one of 12 v = div(F), 12 v = 1G, 12 v = 1(div H ), with v = 0 and ∂v/∂n = 0 on ∂ Br . Then for any 1 < q < ∞, the solution v satisfies the corresponding member of

3

∇ v q . kFk L q (Br ) , L (Br )

2

∇ v q . kGk L q (Br ) , L (Br ) k∇vk L q (Br ) . kH k L q (Br ) . For any Br and p > 1, we define Z 1/2 Z 1/4 2 2 E(u)(Br ) ≡ |∇ u| + |∇u|4 , Br Br Z Z 1/ p p M p (u)(Br ) ≡ \ |u − u| where u = \ u, Br Br Z 1/ p D p (u)(Br ) ≡ r p \ |∇u| p . Br

The following is the main technical lemma: Lemma 2.2. Let u ∈ W 2,2 (B, S n ) be a weakly intrinsic biharmonic map. Then for any p1 such that 2 < p1 < 4 and 1/ p0 = 1/ p1 − 1/4 and for any 0 < β < 1, there exists τ < 1/4 and ε > 0 such that if E(u)(B) < ε, then (M p0 (u) + D p1 (u))(Bτ ) < τ β (M p0 (u) + D p1 (u))(B).

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YIN BON KU

Proof. We fix some 1/2 ≤ r < 1 to be chosen later. Let v = u − h where 12 h α = 0 P4 on Br and h α = u α and ∂h α /∂n = ∂u α /∂n on ∂ Br . Write v = i=1 vi such 2 that 1 vi = (Ti terms) on Br and vi = ∂vi /∂n = 0 on ∂ Br for i = 1, 2, 3, 4. By Proposition 2.1 and Lemma 2.1, we get k∇ 3 v1 k L p3 (Br ) + k∇ 2 v2 k L p2 (Br ) + k∇v3 k L p1 (Br ) + k∇ 3 v4 k L p3 (Br )

. |∇u||∇ 2 u||u − c| L p3 (Br ) + |∇u|2 |u − c| L p2 (Br ) + |∇u||u − c| L p1 (Br ) X

1/2

p + |∇u|2 |u − c| (u α ∇u β − u β ∇u α )2 , L 3 (Br ) α,β, j

where 1/ p2 = 1/ p3 − 1/4, 1/ p1 = 1/ p2 − 1/4 and c = (c1 , . . . , cn+1 ). Then by Sobolev imbedding theorem, we get

k∇vk L p1 (Br ) . |∇u||∇ 2 u||u − c| L p3 (B ) + |∇u|2 |u − c| L p2 (B ) r r

2

+ |∇u||u − c| L p1 (Br ) + |∇u| |u − c||∇u| L p3 (Br ) . Using the Hölder inequality, we have, for 1/ p0 = 1/ p1 − 1/4, k∇vk L p1 (Br ) . ku − ck L p0 (Br ) k∇uk L 4 (Br ) k∇ 2 uk L 2 (Br ) + ku − ck L p0 (Br ) k∇uk2L 4 (Br ) + ku − ck L p0 (Br ) k∇uk L 4 (Br ) + k∇uk3L 4 (Br ) ku − ck L p0 (Br ) . Applying the Sobolev imbedding theorem again to the left hand side, we get kvk L p0 (Br ) + k∇vk L p1 (Br )

2

. ∇ 2 u L 2 (Br ) + k∇uk2L 4 (Br ) + k∇uk3L 4 (Br ) + k∇uk L 4 (Br ) × ku − ck L p0 (Br ) . (E 3 (u) + E 2 (u) + E(u))(B) ku − ck L p0 (Br ) . Now, with this key estimate, the proof proceeds exactly the same as in [Chang et al. 1999c]. But we write it down for the sake of completeness. Set c = u and we choose r with 1/2 ≤ r < 1 such that Z 1/ p0 Z 1/ p1 Z 1/ p0 Z 1/ p1 p0 p1 p0 |u − u| + |∇u| . |u − u| + |∇u| p1 . ∂ Br

∂ Br

B

B

Then for any τ with 0 < τ < 1/4 and x ∈ Bτ , the above justifies the second . in Z Z Z 1/ p0 Z 1/ p1 ∂u p0 p1 |∇h(x)| . |u − u| + . |u − u| + |∇u| ∂ Br ∂ Br ∂n B B = (M p0 (u) + D p1 (u))(B).


49

For any τ < 1/4, this in turn justifies the final step in (M p0 (u) + D p1 (u))(Bτ ) Z Z 1/ p0 1/ p1 = τ −4 |u − u| p0 + τ p1 −4 |∇u| p1 Bτ

Bτ

+τ

=τ

−4/ p0

ku − uk

.τ

−4/ p0

ku − h(0)k L p0 (Bτ ) + τ

L p0 (Bτ )

1−4/ p1

k∇uk L p1 (Bτ )

1−4/ p1

k∇uk L p1 (Bτ )

. τ −4/ p0 kvk L p0 (Bτ ) + kh − h(0)k L p0 (Bτ )

+ τ 1−4/ p1 k∇vk L p1 (Bτ ) + k∇hk L p1 (Bτ )

. τ 1−4/ p1 (E 3 (u) + E 2 (u) + E(u))(B) ku − uk L p0 (B) + τ sup |∇h(x)| . τ 1−4/ p1 ε ku − uk L p0 (B) + τ (M p0 (u) + D p1 (u))(B),

x∈Bτ

where E(u)(B) < ε. If we choose τ sufficiently small, and then ε small, we get (M p0 (u) + D p1 (u))(Bτ ) ≤ τ γ (M p0 (u) + D p1 (u))(B).

Proof of Theorem 2.1. Take any point x ∈ B. Suppose Bρ (x) ⊂ B is such that E(u)(Bρ (x)) < ε. By Lemma 2.2, we know that (M p0 (u) + D p1 (u))(Bτρ (x)) < τ γ (M p0 (u) + D p1 (u))(Bτρ (x)). Note that E(u)(Bs (x)) < ε for all s < ρ. So we can apply the Lemma 2.2 iteratively and get (M p0 (u) + D p1 (u))(Bτ j ρ (x)) ≤ τ γ j (M p0 (u) + D p1 (u))(Bτρ (x)) for j ∈ N. From this, it can be shown that D p1 (Bs (y)) ≤ Cs γ for some C > 0, for all y near x, and sufficiently small s > 0; see [Giaquinta 1983]. Then it follows that u is locally Hölder continuous with exponent β = γ /4 in B using Morrey’s condition; again see [Giaquinta 1983]. 3. Higher interior regularity Here we show that a weakly intrinsic biharmonic map u is smooth on B once it is continuous on B, hence completing the proof of interior regularity. 3.1. Two remarks. In fact, we consider a more general class of elliptic system and prove the following theorem: Theorem 3.1. If u is a weak continuous solution of the system 12 u α = f α (x, Du, D 2 u) +

4 X ∂g α i

i=1

∂ xi

(x, Du, D 2 u) on B,

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YIN BON KU

where | f α (x, P, M)| ≤ λ1 (1 + |P|4 + |M|2 ), |giα (x, P, M)| ≤ λ2 (1 + |P|3 + |M|3/2 ), then u ∈ C 2,β (B) for some β ∈ (0, 1). According to classical regularity theory, once the solution is C 2,β (B), it is smooth on B. Since the Euler–Lagrange equation satisfied by u is included in this class, we have the following: Corollary 3.1. If u is a continuous weakly intrinsic biharmonic map on B, then it is smooth on B. Combining this with the result in Section 2, we finally get the main interior regularity theorem: Theorem 3.2. If u ∈ W 2,2 (B, S n ) is a weakly intrinsic biharmonic map, then u ∈ C ∞ (B, S n ). Two remarks: First, to show that u is C 2,β (B) we only need to show that u 1 (x) = (u(r x) − u(0))/c(r ) belongs to C 2,β (B), where c(r ) = ku − u(0)k L ∞ (Br ) + r . We may assume u 1 (x) to be small when u is continuous on B and r is sufficiently small. Then we get (2)

12 u 1 = f˜α (x, Du 1 , D 2 u 1 ) +

4 X ∂ g˜ α i=1

∂ xi

(x, Du 1 , D 2 u 1 ),

where r4 α c(r ) c(r ) f˜α (x, P, M) = f r x, P, 2 M , c(r ) r r 3 r c(r ) c(r ) g˜iα (x, P, M) = giα r x, P, 2 M . c(r ) r r g˜iα

Thus u 1 is a weak continuous solution of the same type of equations with f˜α , and a˜ cdst satisfying the following growth conditions:

(3)

| f˜α (x, P, M)| ≤ λ˜ 1 (1 + µ1 |P|4 + µ1 |M|2 ), |g˜ α (x, P, M)| ≤ λ˜ 2 (1 + µ2 |P|3 + µ2 |M|3/2 ),

where λ˜ 1 = c(r )1/2 λ1 , µ1 = c(r )1/2 , λ˜ 2 = c(r )1/4 λ2 and µ2 = c(r )1/4 . So λ˜ j and µ j for j = 1, 2 can be made arbitrarily small as r is small. This important observation allows us to reduce the proof of Theorem 3.1 to a scaling argument. Second, the theorem holds if we replace 1u by any elliptic systems.


51

3.2. Proof of Theorem 3.1. First of all, we need the following lemma: Lemma 3.1. Suppose v is a weak solution of the Equation (2) satisfying the growth conditions (3). And suppose µ1 kvk L ∞ ≤ δ

(4)

Z

|D 2 v|2 d x +

and

Z

B

|Dv|4 d x

1/2

Z

|v|2 d x ≤ 1.

+

B

B

Then there exists an r0 > 0 such that for r < r0 , (5) r

Z

4

|D (v − h)| d x + r 2

2

4

Br

Z

4

|D(v − h)| d x

1/2

Br

Z

|v − h|2 d x

+ Br

. λ˜ 21 + λ˜ 22 + δ, where h : Br0 → R K is such that 12 h = 0 in Br0 , and h = v and ∂h/∂n = ∂v/∂n on Br0 . Proof. Using the Sobolev inequality and integration by parts, we have r

Z

Z

1/2

Z

+ |v − h|2 d x |D (v − h)| d x + r |D(v − h)| d x Br Br Br Z Z 1/2 Z ≤ r0 4 |D 2 (v − h)|2 d x + r0 4 |D(v − h)|4 d x + |v − h|2 d x

4

2

2

4

4

Br0

Z

Br0

|D 2 (v − h)|2 d x .

.

Z

|1(v − h)|2 d x Br0

Br0

Z

Br0

λ˜ 1 |v − h| + λ˜ 1 µ1 |v − h||Dv|4 + λ˜ 1 µ1 |v − h||D 2 v|2 d x

. Br0

Z

λ˜ 2 |D(v − h)| + λ˜ 2 µ2 |D(v − h)||Dv|3 + λ˜ 2 µ2 |D(v − h)||D 2 v|3/2 d x.

+ Br0

By [Chang et al. 1999a] we have the estimate |(u − h)(x)| . osc(u)(B1 ) + kDuk L 4 (∂ Br0 ) ≤ kuk L ∞ (B1 ) + 1 if we choose r0 > 1/2 such that Z

4

∂ Br0

Z

|Du|4 d x.

|Du| dσ . B

52

YIN BON KU

Using this estimate and the interpolation inequality, we get Z 1 ˜2 2 2 4 2 2 ˜ left side . λ + ε |v − h| + (δ + 1) λ (|Du| + |D u| ) dx 1 1 2 1 Br0 ε1 Z 1 ˜2 2 2 λ + ε |D(v − h)| dx + 2 2 2 Br0 ε2 Z Z 3/2 1 2 4 ˜ +λ2 µ2 ε3 |D(v − h)| d x + 2 |Du|4 d x ε3 Br0 Br0 Z 3/2 Z 1 2 4 ˜ |D 2 u|2 d x . +λ2 µ2 ε3 |D(v − h)| d x + 2 ε3 Br0 Br0 From this, by choosing a suitable ε j , we obtain the required estimate (5).

Using Lemma 3.1, we can prove an important corollary: Corollary 3.2. For any 0 < β < 1 and sufficiently small λ˜ i with µi > 0, there exists 0 < τ < 1/4 such that if v is a weak solution of Equation (2) with growth conditions (3) that satisfies conditions (4), then there exists a second-order polynomial p(x) = 1 2 x Ax + Bx + C such that Z Z Z 1/2 1 1 2 2 4 \ |D (v − p)| d x + 4 \ |D(v − p)| d x + 4 \ |v − p|2 d x ≤ τ 2β . τ τ Bτ Bτ Bτ Also |A| + |B| + |C| ≤ C0 , where C0 is a universal constant. Proof. Let h be the biharmonic vector in the previous lemma, then Z Z 1/2 Z 1/4 2 (6) khkC 3 (B1/4 ) . (|u| + |Du|)dσ . |u| d x + |Du|4 d x ≤ C0 . ∂ Br0

B1

B1

Let p(x) be the second-order Taylor polynomial of h at 0, that is, let p(x) = 1 2 2 x D h(0)x + Dh(0)x + h(0). By Lemma 3.1, we have, for τ < 1/4, Z Z Z 1/2 1 1 2 2 4 + 4 \ |v − p|2 d x \ |D (v − p)| d x + 4 \ |D(v − p)| d x τ τ Bτ Bτ Bτ Z Z 1 Z 1/2 1 2 2 4 ≤ \ |D (v − h)| d x + 4 \ |D(v − h)| d x + 4 \ |v − h|2 d x τ τ Bτ Bτ Bτ Z Z 1 Z 1/2 1 + \ |D 2 (h − p)|2 d x + 4 \ |D(h − p)|4 d x + 4 \ |h − p|2 d x τ τ Bτ Bτ Bτ ≤ Cτ −8 (λ˜ 21 + λ˜ 22 + δ) + sup|D 3 h|τ 2 ≤ Cτ −8 (λ˜ 21 + λ˜ 22 + δ) + C0 τ 2

(by Lemma 3.1)

(by (6)).


53

Now, first take τ small such that the second term is less than or equal to τ 2β /2, and then take λ˜ j , µ j small (so that δ is also small) such that the rest is bounded by τ 2β /2. Then the result follows. Proof of Theorem 3.1. First we prove this claim: There exists C > 0, 0 < λ˜ i , µi < 1, and ε0 > 0 such that if |u| ≤ 1 and u is a weak solution of Equation (2) with growth condition (3) and λ˜ i has µi ≤ ε0 , then for each k ∈ N there is a second order polynomial pk (x) = 21 x Ak x + Bk x + Ck such that Z Z Z 1/2 1 1 4 2 2 (7) \ |D (u − pk )| d x + 4k \ |D(u − pk )| d x + 4k \ |u − pk |2 d x τ τ Bτ k Bτ k Bτ k ≤ τ 2βk and |Ak | + |Bk | + |Ck | ≤ C, where C is a universal constant. We prove this claim by induction on k. Using Corollary 3.2, the case k = 1 is true. To verify the inductive step, assume the claim is true for k and define wk (x) =

(u − pk )(r k x) . r (2+β)k

Then we get 1

2

wkα

α

= F (x, Dwk , D wk ) + 2

4 X ∂G α i=1

∂ xi

(x, Dwk , D 2 wk ),

where F α (x, P, M) = τ (2−β)k f˜α (τ k x, Dpk (τ k x) + τ (1+β)k P, D 2 pk (τ k x) + τ βk M) +τ (2−β)k (Dcd a˜ cdst )(Dst pk )(τ k x), G iα (x, P, M) = τ (1−β)k g˜iα (τ k x, Dpk (τ k x) + τ (1+β)k P, D 2 pk (τ k x) + τ βk M). Next we check the growth conditions (3): |F α (x, P, M)| ≤ τ (2−β)k λ˜ 1 1 + 8µ1 (C4 + τ 4(1+β)k |P|4 ) + 2µ1 (C2 + τ 2βk |M|2 ) + ε C ≤ λ˜ (1 + 8µ1 τ (6+3β)k |P|4 + 2µ1r (2+β)k |M|2 ), |G α (x, P, M)| ≤ τ (1−β)k λ˜ 2 1 + 4µ2 (C3 + τ 3(1+β)k |P|3 ) + 2µ2 (C3/2 + τ 3βk/2 |M|3/2 ) ≤ λ˜ 2 1 + 4µ2 τ (4+2β)k |P|3 + 2µ2 τ (1+βk/2) |M|3/2 . for λ˜ j , µ j and τ sufficiently small. Now we verify the conditions (4) for wk : 2µ1 τ (2+β)k kwk k L ∞ (B1 ) = 2µ1 ku − pk k L ∞ (Bτ k ) ≤ 2µ1 (1 + C) ≤ δ,

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YIN BON KU

if µ1 is initially chosen to be small. Also, we have Z Z 1/2 Z 2 2 4 |D wk | d x + |Dwk | d x + |wk |2 d x B

B

B

Z Z (u − p )(τ k x) 4 1/2 k k 2 (u − pk )(τ x) 2 = D dx + dx D (2+β)k (2+β)k τ τ B B Z k (u − pk )(τ x) 2 + dx τ (2+β)k B Z Z 1/2 1 1 2 2 = 2βk \ |D (u − pk )| d x + 4k \ |D(u − pk )|4 d x τ τ Bτ k Bτ k Z 1 2 + 4k \ |u − pk | d x τ Bτ k ≤ 1, by the induction hypothesis. So conditions (4) for wk are satisfied. Therefore, we can apply the Corollary 3.2 to wk , that is, there exists a second order polynomial q(x) = 12 x Ax + Bx + C such that Z Z Z 1/2 1 1 2 2 4 \ |D (wk − q)| d x + 4 \ |D(wk − q)| d x + 4 \ |wk − q|2 d x ≤ τ 2β τ τ Bτ Bτ Bτ and |A| + |B| + |C| ≤ C0 . Then define pk+1 (x) = pk (x) + τ (2+β)k q(x/τ k ). By a change of variable, we get Z Z 1/2 1 2 2 |D (u − pk+1 )| + 4(k+1) \ \ |D(u − pk+1 )|4 d x τ Br k+1 Bτ k+1 Z 1 + 4(k+1) \ |u − pk+1 |2 d x ≤ τ 2(k+1)β . τ Br k+1 This proves the inequality (7) for k + 1. Now, it remains to show that |Ak+1 | + |Bk+1 | + |Ck+1 | ≤ C. Initially, we set C = (3C0 )/(1 − 4−β ). From the induction step, we know that for j ≤ k, we have |A j+1 | ≤ |A j | + τ β j C0 , |B j+1 | ≤ |B j | + τ (1+β) j C0 , |C j+1 | ≤ |C j | + τ (2+β) j C0 . This implies |A j+1 | + |B j+1 | + |C j+1 | ≤ |A j | + |B j | + |C j | + 3τ βk C0 . Hence we have |Ak+1 | + |Bk+1 | + |Ck+1 | ≤

3C0 ≤ C. 1 − τβ


55

This complete that proof of the claim for k + 1. Now, similarly to the proof of Theorem 2.1, this result implies that u ∈ C 2,β (B), hence finishing the proof of Theorem 3.1. 4. Boundary regularity Here we will investigate the boundary regularity of weakly intrinsic biharmonic maps u. The main is this: Theorem 4.1. Suppose u ∈ W 2,2 (B 4 , S n ) is a weakly intrinsic biharmonic map such that u|∂ B ∈ C l,β (∂ B, S n ), and ∂u/∂n|∂ B ∈ C l−1,β (∂ B, S n ) for l ∈ N and β ∈ (0, 1). Then u ∈ C l,β (B, S n ). Since the interior regularity has already been established in previous section, we concentrate on the neighborhood of the boundary ∂ B. Without losing generality, n k+1 , where + is the upper-half ball we may assume that u : (+ r 1 , g) → S ⊂ R of radius r , that is, r+ = {(x, t) ∈ R4 | t ≥ 0, |x|2 + t 2 < r }. Then, the Dirichlet boundary conditions become (8)

u(x, 0) ∈ C l,β (01 , S n ) and

∂u (x, 0) ∈ C l−1,β (01 , S n ), ∂n

where 01 is the flat part of ∂+ 1. 4.1. Boundary C 0,β regularity. To prove the main theorem, we first need to prove the boundary C 0,β regularity of u, a consequence of this theorem: n Theorem 4.2. Let u ∈ W 2,2 (+ 1 , S ) be a weakly intrinsic biharmonic map sat0,β n isfying (8). Then u ∈ C (U, S ), where U is a neighborhood of 0s for some s ∈ (0, 1) in + 1.

Proof. First, for any r > 0 we define Z 1/ p M p (u)(r+ ) = \ |u − u(0)| p r+

and

Z D p (u)(r+ ) = r p \

|∇u| p r+

1/ p

.

Suppose 1/2 < r1 < 1 and 0 < τ < r1 /4, with both τ and r1 to be chosen later. Let h 1 : r+1 → Rn+1 be such that 12 h 1 = 0 in r+1 and h 1 = u and ∂h 1 /∂n = ∂u/∂n on ∂r+1 . For p0 and p1 as in Section 2, we have Z Z 1/ p0 1/ p1 + + p0 p1 + τ \ |∇u| p1 M p0 (u)(τ ) + D p1 (u)(τ ) = \ |u − u(0)| + + τ τ Z 1/ p0 1/ p1 Z ≤ \ |u − h 1 | p0 + τ p1 \ |∇(u − h 1 )| p1 + + τ τ Z 1/ p1 Z 1/ p0 + τ p1 \ |∇h 1 | p1 . + \ |h 1 − h 1 (0)| p0 + τ

+ τ

56

YIN BON KU

Similarly to Section 2, we get the key estimate on r+1 : Z Z p1 / p0 |∇(u − h 1 )| p1 . (E 3 + E 2 + E)(r+1 ) p1 |u − u(0)| p0 . r+1

r+1

Apply this and the Sobolev inequality, we get Z 1/ p0 1/ p1 Z + p0 p1 p1 M p0 (u)(+ )+ D (u)( ) . + \ |h −h (0)| + τ \ |∇h | 1 1 1 p1 τ τ + + τ τ Z 1/ p0 1 p0 ) |u − u(0)| . + 4/ p0 (E 3 (u) + E 2 (u) + E(u))(+ 1 τ + 1 Now we apply the above inequality to u(τ k−1 x) for k = 2, 3, . . .. Then by a change of variable, we get Z Z 1/ p0 1/ p1 + + p0 p1 M p0 (u)(τ k )+ D p1 (u)(τ k ) . + \ |h k −h k (0)| + τ \ |∇h k | p1 + + τ τ Z 1/ p0 + −4/ p0 3 2 |+τ (E (u) + E (u) + E(u))(1 ) \ |u − u(0)| p0 , +k−1 τ

where h k : r+k → Rn+1 is such that 12 h k = 0 in r+k , and h k (x) = u(τ k−1 x) and ∂h k (x)/∂n = ∂(u(τ k−1 x))/∂n on ∂r+k , for some rk ∈ (r1 /2, r1 ] to be chosen later. Now define h˜ k (x) = h 1 (τ k−1 x). We have M p0 (u)(+ ) + D p1 (u)(+ ) τk τk .τ

−4/ p0

(E (u) + E (u) + 3

2

Z \

E(u))(+ 1)

|u − u(0)| p0

1/ p0

+k−1

τ Z Z 1/ p0 1/ p1 + \ |h k − h˜ k | p0 + τ p1 \ |∇(h k − h˜ k )| p1 + + τ τ Z Z 1/ p0 1/ p1 + \ |h˜ k − h˜ k (0)| p0 + τ p1 \ |∇ h˜ k | p1 + + τ τ Z 1/ p0 p0 . τ −4/ p0 (E 3 (u) + E 2 (u) + E(u))(+ ) \ |u − u(0)| 1

+k−1

Z + \ +k τ

Z τ 1/ p1 1/ p0 |h 1 − h 1 (0)| p0 + τ kp1 \ |∇h 1 | p1 + τ sup|∇φk | +k τ

+ τ

+ . τ −4/ p0 (E 3 (u) + E 2 (u) + E(u))(+ 1 )M p0 (u)(τ k−1 )

+ M p0 (h 1 )(+ ) + D p1 (h 1 )(+ ) + τ sup|∇φk |, τk τk + τ

where φk = h k − h˜ k . Note that 12 φk = 0 in r+k and φk = ∂φk /∂n = 0 on 0rk . Therefore, by Schauder theory (see [Agmon et al. 1959]), we know that φk is


57

smooth on + τ . Moreover, let G be the Green function of 12 on r+k satisfying Dirichlet boundary conditions. By Green’s identity and that φk = ∂φk /∂n = 0 on 0rk , we get Z ∂(1G) ∂φk (x, y)φk (y) − 1G(x, y) (y) dσ (y). φk (x) = ∂n ∂n ∂r+k \0rk So for x ∈ + τ , we have the estimate Z ∂φ k |φk | + sup|∇φk | . dσ + ∂n + ∂rk \0rk τ Z = |(u − h 1 )(τ k−1 x)| + |∇((u − h 1 )(τ k−1 x))| dσ (x). ∂r+k \0rk

Now we choose rk such that Z Z |(u − h 1 )(τ k−1 x)|dσ (x) . ∂r+k \0rk

Z ∂r+k \0rk

|∇((u − h 1 )(τ k−1 x))|dσ (x) .

+ 1

Z + 1

|(u − h 1 )(τ k−1 x)|, |∇((u − h 1 )(τ k−1 x))|.

Applying these estimates and the Hölder inequality, we get Z 1/ p0 Z 1/ p1 sup|∇φk | . |(u − h 1 )(τ k−1 x)| p0 + |∇((u − h 1 )(τ k−1 x))| p1 + 1

+ τ

.

Z

+ 1

|u(τ k−1 x) − u(0)| p0

1/ p0

+

Z

+ 1

+

Z + 1

|∇(u(τ k−1 x))| p1

1/ p1

+ 1

|h 1 (τ

k−1

x) − h 1 (0)|

p0

1/ p0

+

Z + 1

|∇(h 1 (τ k−1 x))| p1

. M p0 (u)(+ ) + D p1 (u)(+ ) τ k−1 τ k−1 + M p0 (h 1 )(+ ) + D p1 (h 1 )(+ ). τ k−1 τ k−1 Therefore, we have M p0 (u)(+ ) + D p1 (u)(+ ) τk τk + . τ −4/ p0 (E 3 (u) + E 2 (u) + E(u))(+ 1 )M p0 (u)(τ k ) + τ M p0 (u)(+ ) + D p1 (u)(+ ) τ k−1 τ k−1

+ M p0 (h 1 )(+ ) + D p1 (h 1 )(+ ) τ k−1 τ k−1 + M p0 (h 1 )(+ ) + D p1 (h 1 )(+ ) for k = 1, 2, . . . . τk τk

1/ p1

58

YIN BON KU

By the definition of h 1 and the boundary assumption on u, we can deduce from Schauder theory that h 1 ∈ C 1,β (+ τ , S n ), and so for k ∈ N, we have ) ≤ Cτ kβ ) + D p1 (h 1 )(+ M p0 (h 1 )(+ τk τk for some constant C > 0 independent of k and some sufficiently small τ . Now first choose τ small. Then for sufficiently small E(u)(+ 1 ) for k ∈ N, we have )≤ ) + D p1 (u)(+ M p0 (u)(+ τk τk

τβ + (k−1)β . M p0 (u)(+ k−1 ) + D p1 (u)(τ k−1 ) + Cτ τ 2

Now we can apply this inequality iteratively and get (9)

+ M p0 (u)(+ ) + D p1 (u)(+ ) ≤ τ kβ M p0 (u)(+ 1 ) + D p1 (u)(1 ) τk τk C C + + 2 + · · · . τ kβ 2 2

for all k ∈ N. In fact, we can apply the argument to all x ∈ 0s for some s ∈ (0, 1) and obtain the estimate (9) for x. Then by a standard argument we can prove that u ∈ C 0,β (U, S n ), where U is a neighborhood of 0s in + 1 4.2. Proof of Theorem 4.1 for l = 1. This case is in fact a consequence of this theorem: n Theorem 4.3. Let u ∈ W 2,2 (+ 1 , S ) be a weakly intrinsic biharmonic map satis1,β fying (8) for l = 1. Then u ∈ C (U, S n ), where U is a neighborhood of 0s for some s ∈ (0, 1) in + 1.

First, for any r > 0, we define M 0p0 (u)(r+ )

≡

1

Z \

|u − u(0) − ∇u(0)x| p0

1/ p0

r p0 r+ Z 1/ p1 D 0p1 (u)(r+ ) ≡ \ |∇u − ∇u(0)| p1 .

,

r+

We have to rewrite the right side of the Euler–Lagrange equation again so as to obtain the right estimate. First, from the proof of Proposition 2.1 and [Chang et al. 1999c], we observe that 12 u α = Tl terms + 1(u α |∇u|2 )

for l = 1, 2, 4.

Now we rewrite each of these terms in the following way:


59

Type T1 terms: (IA) = div(∇u α 1u β (u β − cβ ) = div((∇u α − a α )1u β (u β − cβ ) + div(a α 1u β (u β − cβ )), (IB) = div(h∇u β , ∇∇u β i(u β − cβ )) = div((u β − cβ )h(∇u α − a α ), ∇∇u β i) + div((u β − cβ )ha β , ∇∇u β i). Type T2 terms: (IIA) = 1((u β − cβ )|∇u β |2 ) = 1((u β − cβ )h∇u β , ∇u α − a α i) + 1((u β − cβ )h∇u β , a β i), (IIB) = 1(u α (u β − cβ )1u β ) = 1 div(u α (u β − cβ )(∇u α − a α )) − 1(u α h∇u α − a α , ∇u β i) − 1((u β − cβ )h∇u α , ∇u β − a β i). Term of the form 1((u β − cβ )1u α ) do not appear. Type T4 terms: (IV) = div(|∇u|2 (u α ∇u β − u β ∇u α )(u β − cβ )) = div(|∇u|2 u α (∇u β − a β )(u β − cβ )) + div(a β |∇u|2 u α (u β − cβ )) − div(|∇u|2 u β (∇u α − a α )(u β − cβ )) + div(a β |∇u|2 u α (u β − cβ )) . 1(u α |∇u|2 ) terms: (V) = 1(u α |∇u|2 ) = div(∇(u α |∇u|2 )) = div(∇u α |∇u|2 ) + 2 div(u α h∇∇u β , ∇u β i) = 1((u β − cβ )|∇u β |2 ) − 2 div((u β − cβ )h∇u β , ∇∇u β i) + 21(u α h∇u β − a β , ∇u β i) − 2 div(∇u α h∇u β − a β , ∇u β i) − 2 div(u α h∇u β − a β , ∇∇u β i). Then (V) = (IIA) term + (IB) term + 21(u α h∇u β − a β , ∇u β i) − 2 div(∇u α h∇u β − a β , ∇u β i) − 2 div(u α h∇u β − a β , ∇∇u β i), P4 β where a β = i=1 ai ∂/∂ xi is any constant vector field and cβ is any constant. Now we are ready to prove this technical lemma:

60

YIN BON KU

Lemma 4.1. For any r ∈ (0, 1), the estimate Z Z p1 2 + p1 p1 |∇(u − h)| . (E + E)(u)(r ) + max|u − c| r+

r+

|∇u − a| p1 Z + |a| |u − c| p1

r+

r+

holds on r+ , where h : r+ → Rk+1 is such that 12 h = 0 in r+ and such that h = u and ∂h/∂n = ∂u/∂n on ∂r+ . Proof. Using Lemma 2.1 in Section 2 and the Hölder inequality, we get k∇(u − h)k L p1 (r+ ) . (IA)0 + (IB)0 + (IIA)0 + (IIB)0 + (IV)0 + (V)0 terms, where (IA)0 = k1uk L 2 (r+ ) k∇u − ak L p1 (r+ ) + |a|k1uk L 2 (r+ ) ku − ck L p1 (r+ ) , (IB)0 = k∇ 2 uk L 2 (r+ ) k∇u − ak L p1 (r+ ) + |a|k∇ 2 uk L 2 (r+ ) ku − ck L p1 (r+ ) , (IIA)0 = k∇uk L 4 (r+ ) k∇u − ak L p1 (r+ ) + |a|k∇uk L 4 (r+ ) ku − ck L p1 (r+ ) , (IIB)0 = (max|u − c|)k∇u − ak L p1 (r+ ) + k∇uk2L 4 (+ ) k∇u − ak L p1 (r+ ) , r+

r

(IV)0 = k∇uk2L 4 (+ ) k∇u − ak L p1 (r+ ) + |a|k∇uk2L 4 (+ ) ku − ck L p1 (r+ ) , r

0

0

r

0

(V) = (IB) + (IIA)

+ k∇uk2L 4 (+ ) + k∇uk L 4 (r+ ) + k∇ 2 uk L 2 (r+ ) k∇u − ak L p1 (r+ ) . r

After grouping terms, it is easy to obtain the required estimate.

Proof of Theorem 4.3. Suppose 1/2 < r1 < 1 and 0 < τ < r1 /4, but both τ and r1 are otherwise to be chosen later. Define h 1 as in previous section. By the Sobolev inequality, we have 0 + M 0p0 (u)(+ τ ) + D p1 (u)(τ ) Z 1/ p1 −4/ p1 0 + .τ \ |∇(u − h 1 )| p1 + M 0p0 (h 1 )(+ τ ) + D p1 (h 1 )(τ ) r+1

. τ −4/ p1 (E 2 + E)(u)(+ 1 ) + max|u − u(0)|

+ 1

+ 0 + 0 + × D 0p1 (u)(+ 1 ) + |∇u(0)|M p1 (u)(1 ) + M p0 (h 1 )(τ ) + D p1 (h 1 )(τ ). The last inequality follows from Lemma 4.1 by setting cα = u α (0) and a β = ∇u β (0). Now we apply the above inequality to u(τ k−1 x) for k = 2, 3, . . ., and


61

then by a change of variable, we get M 0p0 (u)(+ ) + D 0p1 (u)(+ ) τk τk . τ −4/ p0 (E 2 + E)(u)(+ 1 ) + max |u − u(0)|

+k−1 τ

+

1 τ k−1

+ × D 0p1 (u)(+ ) + |∇u(0)|M (u)( ) p k−1 k−1 1 τ τ 0 + 0 + M p0 (h k )(τ ) + D p1 (h k )(τ ) ,

where h k is defined on r+k in previous section and rk is to be chosen later. Repeating the proof method of Theorem 4.2, we consider h˜ k (x) = h 1 (τ k−1 x). Then we have 0 + M 0p0 (h k )(+ τ ) + D p1 (h k )(τ ) 0 + 0 ˜ + 0 ˜ + ≤ M 0p0 (φk )(+ τ ) + D p1 (φk )(τ ) + M p0 (h 1 )(τ ) + D p1 (h 1 )(τ ) ≤ τ sup|∇ 2 φk | + M 0p0 (h 1 )(+ ) + D 0p1 (h 1 )(+ ) τ k−1 , τk τk + τ

where φk = h k − h˜ k . Again note that by Schauder theory, we know that φk is smooth on + τ , and so ∇ 2 φk is well defined. As before, by a Green function argument, we have the estimate 1 sup|∇ 2 φk | . M 0p0 (u)(+ ) + D 0p1 (u)(+ ) τ k−1 τ k−1 τ k−1 +τ + M 0p0 (h 1 )(+ ) + D 0p1 (h 1 )(+ ). τ k−1 τ k−1 Combining these results, we get M 0p0 (u)(+ ) + D 0p1 (u)(+ ) τk τk .

1 τ 4/ p0

(E 2 + E)(u)(+ 1 ) + max |u − u(0)|

+k−1 τ

+ × D 0p1 (u)(+ ) + |∇u(0)|M (u)( ) p k−1 k−1 1 τ τ + τ M 0p0 (u)(+ ) + D 0p1 (u)(+ ) + M 0p0 (h 1 )(+ ) τ k−1 τ k−1 τ k−1 + D 0p1 (h 1 )(+ ) + M 0p0 (h 1 )(+ ) + D 0p1 (h 1 )(+ ). τk τk τ k−1 By Schauder theory, we know that h 1 ∈ C 1,β (+ τ ), and we know by Theorem 4.2 that u ∈ C 0,β (+ τ ). Therefore, we have M 0p0 (h 1 )(+ ) + D 0p1 (h 1 )(+ ) . τ βk , τk τk

M p1 (u)(+ ) . τ β(k−1) , τ k−1 max |u − u(0)| . τ β(k−1)

+k−1 τ

62

YIN BON KU

for k = 2, 3, . . . and τ sufficiently small. With these estimates, we first choose k0 ∈ N such that (k0 − 1)β − 4/ p0 ≥ 1, then choose τ small, and finally, for E(u)(+ 1 ) sufficiently small, we get M 0p0 (u)(+ ) + D 0p1 (u)(+ )≤ τk τk

τβ + 0 (k−1)β M 0p0 (u)(+ k−1 ) + D p1 (u)(τ k−1 ) + Cτ τ 2

for some constant C > 0 independent of k and k ≥ k0 . Then iteratively applying the above inequality we get τ kβ M 0p0 (u)(+ (10) M 0p0 (u)(+ ) + D 0p1 (u)(+ )≤ ) + D 0p1 (u)(+ ) τk τk τ k0 τ k0 2 C C + C + + 2 + · · · . τ kβ for k ≥ k0 . 2 2 Again, as in the proof of Theorem 2.1, we can apply the argument to all x ∈ 0s and obtain the estimate (10) for x. Then by a standard argument it can be shown that u ∈ C 1,β (U, S n ). 4.3. Proof of Theorem 4.1 for l ≥ 2. Again, by standard regularity theory, it suffices to prove the case l = 2. As in Section 3, we consider a larger class of elliptic systems. In this section, we will prove this: Theorem 4.4. Suppose u ∈ C 1,β (+ 1 , S n ) is a weak solution on + 1 of the elliptic system 4 X ∂giα 12 u α = f α (x, Du, D 2 u) + (x, Du, D 2 u) ∂ xi i=1 with growth conditions | f α (x, P, M)| ≤ λ1 (1 + |P|4 + |M|2 ),

(11)

|giα (x, P, M)| ≤ λ2 (1 + |P|3 + |M|3/2 )

and Dirichlet boundary data satisfying (8) for l = 2. Then u ∈ C 2,β (U, S n ), where U is a neighborhood of 0s in + 1 for some s ∈ (0, 1). Since the Euler–Lagrange equation of the intrinsic biharmonic map u belongs to this class of elliptic system and, by the previous section, we already know that u ∈ C 1,β (+ 1 ), we see Theorem 4.4 implies Theorem 4.1. As in Section 3, to show that u ∈ C 2,β (U), it suffices to show that u 1 (x) = (u(r x)−u(0))/c(r, K ) belongs to C 2,β (U), where c(r, K ) = K (ku − u(0)k L ∞ (Br ) + r ) for some K > 1 and r > 0. Since u is continuous, c(r, K ) becomes arbitrarily small as r → 0. Therefore, by a computation in Section 3, we know u 1 satisfies the same type of elliptic system (12)

2 α

1 u = f˜α (x, Du, D 2 u) +

4 X ∂ g˜ α i

i=1

∂ xi

(x, Du, D 2 u) in + 1


63

with growth conditions | f˜α (x, P, M)| ≤ λ˜ 1 (1 + µ1 |P|4 + µ1 |M|2 ),

(13)

|g˜ α (x, P, M)| ≤ λ˜ 2 (1 + µ2 |P|3 + µ2 |M|3/2 )

where λ˜ 1 = c(r, K )1/2 λ1 , µ1 = c(r, K )1/2 , λ˜ 2 = c(r )1/4 λ2 , and µ2 = c(r, K )1/4 . So λ˜ j and µ j for j = 1, 2 can be made arbitrarily small as r goes to zero. Also note we can assume |u|C 1,β (+ 1 ) , |u|C 2,β (01 ) , and |∂u/∂n|C 1,β (01 ) to be very small if we fix a large enough K . To prove Theorem 4.4, we need a lemma. Lemma 4.2. Suppose v is a weak solution of Equation (12) with growth conditions (13) and Dirichlet boundary data satisfying (8) for l = 2. Also suppose

(14)

Z + 1

|D 2 v|2 d x +

Z + 1

µ1 (|v − h| L ∞ (r+ ) ) ≤ δ, 0 1/2 Z 4 2 |Dv| d x + |v| d x ≤ 1 + 1

for some r0 ∈ (0, 1] and h : r+0 → R K such that 12 h = 0 in r+0 and such that h = v and ∂h/∂n = ∂v/∂n on ∂r+0 . Then for 0 < r < r0 , Z Z 1/2 Z |D 2 (v − k)|2 d x + r 4 |D(v − k)|4 d x + |v − k|2 d x . λ˜ 21 + λ˜ 22 + δ. r4 Br

Br

Br

The proof of Lemma 4.2 is similar to that of Lemma 3.1 and is therefore omitted. Proof of Theorem 4.4. First let w0 = u, p0 = 0, and h 0 = h where h : r+0 → R K +1 is such that 12 h = 0 in r+0 and such that h = u and ∂h/∂n = ∂u/∂n on ∂r+0 for some r0 ∈ (0, 1), to be chosen later. Let τ ∈ (0, r0 ) also to be chosen later. For k ∈ N, we define (u − pk )(τ k x) , wk = τ (2+β)k where pk (x) = pk−1 (x) + τ (2+β)k qk−1 (x/τ k ) for qk−1 (x) = 21 x D 2 h k−1 (0)x + Dh k−1 (0)x + h k−1 (0) and h k−1 : r+k−1 → Rn+1 such that 12 h k−1 = 0 in r+k−1 and h k−1 = wk−1 , ∂h k−1 /∂n = ∂wk−1 /∂n on ∂r+k−1 for some rk−1 ∈ (r0 /4, r0 /2), also to be chosen later. Notice that by definition h k (0) = 0 and Dh k (0) = 0 for all k ∈ N. So p1 (x) = 1 1 β 2 2 2 x D h(0)x + Dh(0)x + h(0) and pk (x) = pk−1 (x) + 2 τ x D h k−1 (0)x for k ≥ 2. 2 Also, it can be shown that x D h k−1 (0)x = 0 and D(∂h k−1 /∂n)(0)x = 0 for all x ∈ 0rk−1 for k ≥ 2. To prove Theorem 4.4, it suffices to prove that Z Z Z 1/2 1 1 2 2 4 + 4k \ |u− pk |2 d x ≤ τ 2βk \ |D (u− pk )| d x+ 4k \ |D(u− pk )| d x + + τ τ k k +k τ

τ

τ

64

YIN BON KU

for all k ∈ N and |Ak | + |Bk | + |Ck | ≤ C for some constant C independent of k, where pk (x) = 12 x Ak x + Bk x + Ck . We prove this claim by induction on k. First consider when k = 1. By the discussion at the beginning of this section, we may assume without loss of generality that Z Z 1/2 Z 4 2 |u| d x + |Du| d x + |D 2 u|2 d x ≤ 1, + 1

+ 1

+ 1

8µ1 ku − h 0 k L ∞ (r+ ) ≤ δ, 0 ∂u ≤ δ0 |u|C 1,β (+ 1 ) + |u|C 2,β (01 ) + 1,β ∂n C (01 ) for small δ and δ 0 to be chosen later. Then we have Z Z Z 1/2 1 2 2 4 4 \ |D (u − p1 )| d x + 1/τ \ |D(u − p1 )| d x + 4 \ |u − p1 |2 d x + τ +τ + τ τ Z Z 1/2 Z 1 1 4 2 2 ≤ \ |D (u − h 0 )| d x + 4 \ |D(u − h 0 )| d x + 4 \ |u − h 0 |2 d x τ +τ τ +τ + τ Z Z 1/2 1 + \ |D 2 (h 0 − q0 )|2 d x + 4 \ |D(h 0 − q0 )|4 d x τ +τ + τ Z 1 + 4 \ |h 0 − q0 |2 d x τ +τ ≤ C1 τ −8 (λ˜ 21 + λ˜ 22 + δ) + 3[h 0 ]C2 2,β (+ ) τ 2β τ

(by Lemma 4.2)

≤ C1 τ −8 (λ˜ 21 + λ˜ 22 + δ) + C2 δ 0 τ 2β . Let λ˜ i , δ, and δ 0 be small enough that (15)

τ 2β C1 τ −8 (λ˜ 21 + λ˜ 22 + δ) ≤ 2

and

C2 δ 0 τ 2β ≤

τ 2β . 4

Therefore, the claim is true for k = 1. Now assume the claim is true for k. Similarly, we have Z Z Z 1/2 1 1 2 2 4 \ |D (wk − qk )| d x + 4 \ |D(wk − qk )| d x + 4 \ |wk − qk |2 d x + τ τ +τ + τ τ Z Z 1/2 1 ≤ \ |D 2 (wk − h k )|2 d x + 4 \ |D(wk − h k )|4 d x τ +τ + τ Z Z 1/2 1 2 2 + \ |D (h k − qk )| d x + 4 \ |D(h k − qk )|4 d x τ +τ + τ Z Z 1 1 + 4 \ |wk − h k |2 d x + 4 \ |h k − qk |2 d x τ +τ τ +τ


≤τ

−8

h

τ

4

Z

|D (wk − h k )| d x + τ 2

+ τ

2

4

Z + τ

|D(wk − h k )|4 d x

65

1/2

1/2 Z 1 2 2 |D (h k − qk )| d x + 4 \ |D(h k − qk )|4 d x τ +τ + τ Z Z i 1 + |wk − h k |2 d x + 4 \ |h k − qk |2 d x. + τ + τ τ Z +\

To use Lemma 4.2, we must verify conditions (14) for wk . First, by the induction hypothesis, we have Z Z 1/2 Z 2 2 4 |D wk | d x + |Dwk | d x + |wk |2 d x ≤ 1. + 1

+ 1

+ 1

Therefore, the second condition of (14) is satisfied. Second, by the computation in Section 3, the first condition of (14) for wk becomes 8µ1 τ (2+β)k kwk − h k k L ∞ (r+ ) ≤ δ. k

It is easy to see that for k ≥ 0, h k (x) = (hˆ k (x) − pk (τ k x))/τ (2+β)k , where hˆ k : r+k → Rn+1 is such that 12 hˆ k = 0 on r+k and such that hˆ k = u(τ k x) and ∂ hˆ k /∂n = ∂(u(τ k x))/∂n on ∂r+k . So the condition is equivalent to 8µ1 ku(τ k x) − hˆ k (x)k L ∞ (r+ ) ≤ δ. k

By definition, kuk L ∞ (r+ ) ≤ 1 for any k. By the Schauder estimates, we have k

khˆ k (x)k L ∞ (r+ ) . |u|C 1,β (+ 1 ) . δ 0 . k

So by an initial choice of small µ1 and δ 0 , condition 4.3 is satisfied. Now we can apply Lemma 4.2 for wk and get Z Z Z 1/2 1 1 \ |D 2 (wk − qk )|2 d x + 4 \ |D(wk − qk )|4 d x + 4 \ |wk − qk |2 d x τ +τ τ +τ + τ Z ≤ C1 τ −8 (λ˜ 21 + λ˜ 22 + δ) + \ |D 2 (h k − qk )|2 d x + τ

+

Z Z 1/2 1 1 4 + \ |D(h − q )| d x \ |h k − qk |2 d x k k τ 4 +τ τ 4 +τ

66

YIN BON KU

Z τ 2β ≤ + \ |D 2 (h k − h˜ k − qk )|2 d x 2 + τ Z Z 1/2 1 1 4 + 4 \ |D(h k − h˜ k − qk )| d x + 4 \ |h k − h˜ k − qk |2 d x τ +τ τ +τ Z Z Z 1/2 1 1 4 2˜ 2 ˜ + 4 \ |h˜ k |2 d x, + \ |D h k | d x + 4 \ |D h k | d x τ +τ τ +τ + τ where h˜ k (x) = ((h 0 −q0 )(τ k x))/τ (2+β)k . Define φk = h k − h˜ k . Note that 12 φk = 0 in r+k and φk = ∂φk /∂n = 0 on 0rk . Therefore by Schauder theory, φk is smooth on + τ , and so we have Z Z Z 1/2 1 1 2 2 2 \ |D (wk − qk )| d x + 4 \ |wk − qk | d x + 4 \ |D(wk − qk )|4 d x τ +τ τ +τ + τ ≤ C1 τ −8 (λ˜ 21 + λ˜ 22 + δ) + 3[h 0 ]2 2,β + τ 2β + τ 2 sup|D 3 φk |2 (

C

≤

τ 2β 2

+

τ 2β 4

τ)

+ τ

+ τ 2 sup|D 3 φk |2 . + τ

The first and third term of the last inequality follow from Equation (15). As before, we can estimate |D 3 φk |2 as follows: Z Z ∂φk 4 1/2 3 2 2 sup|D φk | ≤ C3 | dσ |φk | dσ + | ∂r+k \0rk ∂r+k \0rk ∂n + τ Z Z 1/2 Z Z 1/2 2 4 2 4 ≤ C4 |wk | d x + + |h˜ k | d x + |Dwk | d x |D h˜ k | d x + 1

≤ C4 1 + 2[h 0 ]C2 2,β (+

+ 1

+ 1

τ)

+ 1

≤ C4 (1 + 2C2 δ 0 ). Then by an initial choice of small τ , we can assume that τ 2 C4 (1+2C2 δ 0 ) ≤ τ 2β /4. Therefore we get Z Z Z 1/2 1 1 \ |D 2 (wk −qk )|2 d x + 4 \ |D(wk −qk )|4 d x + 4 \ |wk −qk |2 d x ≤ τ 2β τ +τ τ +τ + τ By change of variable, we get Z \ |D 2 (u − pk+1 )|2 d x + +k+1 τ

1 τ 4(k+1)

Z \ +k+1

|D(u − pk+1 )|4 d x

τ

+

1 τ 4(k+1)

Z \ +k+1 τ

1/2

|u − pk+1 |2 d x ≤ τ 2β(k+1) .


67

This finishes the proof for k + 1. Finally, we need to show that |Ak | + |Bk | + |Ck | has a bound that is independent of k. Note that Ck = u(0) and Bk = Du(0) for all k. So it suffices to consider Ak . First, we know that Z ∂φ 4 1/2 Z k 2 2 2 2 2 |D h k (0)| = |D φk (0)| ≤ C4 |φk | dσ + , dσ + + ∂rk \0rk ∂rk \0rk ∂n which is less than or equal to C4 (1 + 2C2 δ 0 ). So |D 2 h k (0)| ≤ C5 for some constant C5 independent of k. The desired k-independence then follows by definition: |Ak | = |D 2 h 0 (0)| + τ β |D 2 h 1 (0)| + τ 2β |D 2 h 2 (0)| + · · · ≤ |D 2 h 0 (0)| +

C5 τ β . 1−τ β

References [Agmon et al. 1959] S. Agmon, A. Douglis, and L. Nirenberg, “Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I”, Comm. Pure Appl. Math. 12 (1959), 623–727. MR 23 #A2610 Zbl 0093.10401 [Chang et al. 1999a] S.-Y. A. Chang, M. J. Gursky, and P. C. Yang, “Regularity of a fourth order nonlinear PDE with critical exponent”, Amer. J. Math. 121:2 (1999), 215–257. MR 2000b:49066 Zbl 0921.35032 [Chang et al. 1999b] S.-Y. A. Chang, L. Wang, and P. C. Yang, “Regularity of harmonic maps”, Comm. Pure Appl. Math. 52:9 (1999), 1099–1111. MR 2000j:58024 Zbl 1044.58019 [Chang et al. 1999c] S.-Y. A. Chang, L. Wang, and P. C. Yang, “A regularity theory of biharmonic maps”, Comm. Pure Appl. Math. 52:9 (1999), 1113–1137. MR 2000j:58025 Zbl 0953.58013 [Giaquinta 1983] M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies 105, Princeton University Press, Princeton, NJ, 1983. MR 86b:49003 Zbl 0516.49003 [Hélein 1991] F. Hélein, “Régularité des applications faiblement harmoniques entre une surface et une variété riemannienne”, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), 591–596. MR 92e:58055 Zbl 0728.35015 [Lamm and Riviére ≥ 2008] T. Lamm and T. Riviére, “Conservation laws for fourth order systems in four dimensions”, Comm. Partial Differential Equations. To appear. [Qing 1993] J. Qing, “Boundary regularity of weakly harmonic maps from surfaces”, J. Funct. Anal. 114:2 (1993), 458–466. MR 94h:58065 Zbl 0785.53048 [Rivière 1995] T. Rivière, “Everywhere discontinuous harmonic maps into spheres”, Acta Math. 175:2 (1995), 197–226. MR 96k:58059 Zbl 0898.58011 [Wang 2004] C. Wang, “Biharmonic maps from R4 into a Riemannian manifold”, Math. Z. 247:1 (2004), 65–87. MR 2005c:58030 Zbl 1064.58016 Received February 5, 2007. Revised May 3, 2007. Y IN B ON K U D EPARTMENT OF M ATHEMATICS T HE H ONG KONG U NIVERSITY OF S CIENCE AND T ECHNOLOGY C LEAR WATER BAY, KOWLOON H ONG KONG [email protected]


ON TWO NOTIONS OF SEMISTABILITY M ARIO M AICAN We show that certain semistable sheaves on the projective plane with linear Hilbert polynomial are cokernels of semistable morphisms of decomposable bundles. We exhibit certain locally closed subvarieties or open dense subsets of moduli spaces of semistable sheaves as quotients modulo nonreductive groups. These subvarieties are defined by cohomological conditions. We find isomorphisms between such subvarieties given by sending a sheaf to its dual.

1. Introduction The notion of a (Gieseker) semistable sheaf is well-established in the literature and allows one to construct moduli spaces of sheaves with fixed Hilbert polynomial on a projective variety. The construction, carried out in [Simpson 1994a; 1994b], relies on the existence theorems from geometric invariant theory, more precisely, it is shown that the moduli space occurs as the quotient of a certain set of semistable points of a quotient scheme modulo a reductive algebraic group. To get a semistable quotient from a semistable sheaf F we need to express F as a quotient m O(−d) → F → 0 with large m and d. In general this procedure is quite abstract and of little use for the purposes of describing concretely the geometry of the moduli space. Another approach for studying moduli spaces uses monads. Let MP2 (r, c1 , c2 ) be the moduli space of semistable (in the sense of Mumford and Takemoto) torsionfree sheaves on P2 of rank r and Chern classes c1 , c2 . Assume that there exist locally free sheaves E1 , E2 , E3 on P2 such that each F giving a point in MP2 (r, c1 , c2 ) is the cohomology of a monad 0 → E1 → E2 → E3 → 0. The space W of monads is acted upon in an obvious manner by the algebraic group G = Aut E1 × Aut E2 × Aut E3 . Two fundamental questions now arise: Is there a semistability notion for W such that a monad is semistable precisely if its MSC2000: 14F05, 14L24, 00A05, 14D20, 14D22. Keywords: moduli spaces, nonreductive groups, sheaves on the projective plane, semistable sheaves, sheaves of dimension one. 69

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MARIO MAICAN

cohomology is semistable? And is MP2 (r, c1 , c2 ) a good quotient of the set W ss of semistable monads modulo G ? The description of MP2 (2, c1 , c2 ) as a good quotient was done in [Barth 1977] for c1 even and in [Hulek 1979] for c1 odd. In [Chang 1983] it was shown that a generic stable bundle on P3 of rank 2, Chern classes c1 = 0, c2 = 4 and α-invariant 1 is the cohomology of a self-dual monad. Drézet [1987] described as quotients those MP2 (r, c1 , c2 ) for which 1 = δ. He takes E3 = 0 and E1 , E2 direct sums of certain exceptional bundles. In all these instances the group G was reductive. Quotients by nonreductive G were considered in [Drézet 1991], which studies MP2 (r, c1 , c2 ) of “faible hauteur”. Again E3 = 0, so Drézet is able to express each semistable bundle as the cokernel of a semistable morphism. A notion of semistability for complexes of morphisms of sheaves modulo nonreductive groups was proposed in [Drézet 1991; 1998; Drézet and Trautmann 2003]. We briefly explain the case of morphisms of sheaves. (In this paper we will not need the notion of a semistable complex of length 3 or more.) Drézet and Trautmann consider sheaves E1 and E2 on Pn which are direct sums of simple sheaves, e.g., direct sums of line bundles. Thus Aut E1 × Aut E2 is nonreductive if E1 or E2 has more than one kind of simple sheaf in its decomposition. This group acts on the vector space W = Hom(E1 , E2 ) and the set of semistable points W ss is defined by means of polarizations which will be not detailed here. We refer to Section 3 for the precise definition. In [Drézet and Trautmann 2003] as well as in [Drézet 2000] it was shown that this notion of semistability quite often leads to a theory similar to the geometric invariant theory. Freiermuth and Trautmann [2004] studied the moduli space of semistable (in the sense of Gieseker) sheaves F on P3 with Euler characteristic 1 and with support curves of multiplicity 3. They showed that each F has a resolution ψ

ϕ

0 → 2O(−3) → O(−1) ⊕ 3O(−2) → O(−1) ⊕ O → F → 0 with ϕ semistable in the sense of [Drézet and Trautmann 2003]. Moreover, the moduli space is a geometric quotient of the parameter space of (ψ, ϕ) modulo the action of the group of automorphisms. In this paper we are interested in semistable sheaves on P2 with linear Hilbert polynomial. Let MP2 (r, χ) denote the moduli space of such sheaves F with fixed multiplicity r and Euler characteristic χ. Motivated by [Freiermuth and Trautmann 2004] we will seek to express F as a cokernel ϕ

E1 → E2 → F → 0

with E1 and E2 direct sums of line bundles and ϕ semistable in the sense of Drézet and Trautmann. We carry this out in Sections 4–6 for sheaves satisfying certain

ON TWO NOTIONS OF SEMISTABILITY

71

cohomological conditions. The picture we provide is far from complete because we do not have a full list of resolutions for all F giving a point in MP2 (r, χ) even in the case r = 4 (the cases r = 1, 2 are trivial while the case r = 3 is completely understood). Our cohomological conditions define locally closed subvarieties in MP2 (r, χ) and in Section 7 we address the question whether these subvarieties are good or geometric quotients of the sets of semistable morphisms ϕ modulo the canonical action of the group of automorphisms. We find that when r , χ are mutually prime, in other words when MP2 (r, χ) is a fine moduli space, we always have geometric quotients. If the moduli space is not fine the problem is more complicated and we can answer it only in some cases. In Section 8 we compute the codimensions of all locally closed subsets of MP2 (r, χ) under investigation. In Section 9 we prove a general duality result. The dual of a sheaf F giving a point in MP2 (r, χ) is F D = Ext 1 (F, 2 )(1). Applying the map F → F D to a locally closed subset X in MP2 (r, χ) we get a locally closed subset in MP2 (r, r −χ) denoted X D . At Theorem 9.6 we show that under certain conditions X and X D are isomorphic. In particular, this is true for all sets X under investigation in this paper. Our theorem is inspired from the result present in [Freiermuth 2000], that MP2 (r, χ) and MP2 (r, r −χ) are birational if gcd(r, χ) = 1. We show that this is also true if (r, χ) = (6, 4), (8, 6), or (9, 6). We summarize our results in the tables that follow. The first column of each table, after the header, contains the cohomological conditions defining a locally closed subset X ⊂ MP2 (r, χ). The second shows the codimension of X ; a zero means an open dense subset. Each sheaf F giving a point in X has a resolution of the kind featured in the header. We have more information about these resolutions: the morphisms ϕ of which F is the cokernel form a subset Wo inside the set W ss (G, 3) of morphisms which are semistable with respect to a polarization 3 and to the canonical action of the group G of automorphisms (see Section 3 for the terminology). The third column of our tables contains information about 3 and the forth column says whether X is a quotient of Wo by G. When we write “good” it is understood that the quotient is not geometric; a question mark means we could not prove that a quotient exists. The subset Wo ⊂ W ss (G, 3) is given by the following conditions: for all the blocks different than the last block in the table we require that ϕ be injective and that its scalar entries (regarding it as a matrix) are zero. For the last block we refer to Claims 6.9 and 6.10. MP2 (n +1, n), n ≥ 1 h 0 (F(−1)) = 0

0 → O(−2)⊕(n −1) O(−1) → n O → F → 0 0

0 < λ1
b/a, which shows that C violates the semistability of F.

Remark 6.2. Assume that F is semistable. Then, in case (i), we either have m + 2b − a < b or m + 2b − a ≥ b and all maximal minors of ϕ22 are zero. Similarly, in case (ii), either m < b or m ≥ b and all maximal minors of ϕ22 are zero. This follows from Remark 6.1. Remark 6.3. η is an isomorphism on semistable, ϕ22 cannot have the form  0 0  ... ...   0 0 X Y

global sections. As a consequence, if F is ? ··· .. . ? ··· ? ···

 ? ..  . . ? ?

Indeed, if ϕ22 had the above form, we would get the commutative diagram 2O(−1)

I 0

m O(−1)

[X, Y ]

/O

/ Cx 0 I

ϕ22

/ bO

η12

/0.

/F

Here Cx is the structure sheaf of the point x = (0 : 0 : 1). But the map Cx → F is zero because F does not have zero-dimensional torsion. This shows that η12 has nontrivial kernel. This contradicts the fact that η is an isomorphism on global sections. Claim 6.4. There are no semistable sheaves F on P2 with h 0 (F(−1)) 6= 0, h 1 (F) = 0 and Hilbert polynomial PF (t) = nt + 1,

n≥2

or

PF (t) = nt + 2,

n ≥ 3.

Proof. The case PF (t) = nt +1 follows directly from Remark 6.3 because ϕ22 must have the form [X, Y, ?, · · · , ?]. In the case PF (t) = nt + 2 all 2 × 2-minors of ϕ22 are zero, cf. Remark 6.2. It follows that ϕ22 has the form 0 0 ? ··· ? . X Y ? ··· ?

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MARIO MAICAN

The claim follows from Remark 6.3. Remark 6.5. Let α = (αi j ) be a morphism of sheaves on Pn = P(V ): α : (m + 1) O → m O(l).

Assume that at least one of the maximal minors of α is a nonzero polynomial. Then Ker (α) ' O(−d) where d is an integer satisfying 0 ≤ d ≤ ml. More precisely, let αi , 1 ≤ i ≤ m + 1, denote the minor obtained from α by erasing the i th column. Let β = (β1 , . . . , βm+1 ),

where

βi =

(−1)i αi , g.c.d.(α1 , . . . , αm+1 )

1 ≤ i ≤ m + 1.

Let d be the degree of the entries of β. Then we have the exact sequence β

α

0 → O(−d) → (m + 1) O → m O(l). Claim 6.6. Let F be a semistable sheaf on P2 = P(V ) with h 0 (F(−1)) 6= 0, h 1 (F) = 0 and Hilbert polynomial PF (t) = nt + 3, n ≥ 4. Then h 0 (F(−1)) = 1 and F has a resolution ψ

ϕ

0 → O(−2) → (n − 2) O(−2) ⊕ 3O(−1) → (n − 3) O(−1) ⊕ 3O → F → 0 with ϕ12 = 0, ψ11 = 0, ϕ21 6= 0,    −Y X X ϕ22 ∼  −Z ψ21 ∼  Y  , 0 0 −Z Z

 0 X , Y

ϕ11

ϕ0 0 ? ?

where ϕ 0 is an m × m-matrix with entries in V ∗ , 1 ≤ m ≤ n − 3. Moreover, F is an extension of the form 0 → OC (1) → F → G → 0 where C is a curve of degree d, 4 ≤ d ≤ n, and the map F → G is zero on global sections. If n ≥ 7 then d ≥ 5. Proof. Assume n ≥ 6 so that we are in case (ii). If m ≥ 5 then ϕ22 has the form     ? ? ? 0 ··· 0 ? ? ?  ? ? ? ? · · · ?  =  ? ? ? ϕ0  . ? ? ? ? ? ? ? ··· ? From Remark 6.2 we see that all 2 × 2-minors of ϕ 0 are zero. Since ϕ22 cannot have a zero column it follows that   ? ? ? 0 ··· 0 ϕ22 ∼  ? ? ? 0 · · · 0  . ? ? ? ? ··· ?


103

By virtue of Remark 6.3 this is impossible. Assume now that m = 4. Firstly, we notice that ϕ22 cannot have a zero row because, if 0 ··· 0 , ϕ22 = ϕ0 then, arguing as in Remark 6.1, we get that ϕ 0 has all maximal minors equal to zero hence ϕ 0 has one row identically zero. This, again, contradicts Remark 6.3. Secondly, using the same kind of arguments, we notice that ϕ22 cannot have the form   X 0 0 0  ? ? ? ? . ? ? ? ? Now ϕ22 has nontrivial kernel in ⊕4 V ∗ by hypothesis. No element in the kernel can have the form   X Y     0  0 otherwise we would arrive at a matrix excluded by Remark 6.3:   0 0 ? ? ϕ22 ∼  0 0 ? ?  . −Y X ? ? Performing operations on the columns of ϕ22 we may assume that   X Y    Z 0 is in the kernel of ϕ22 . Performing operations on the rows of ϕ22 we may assume that   −Y X 0 u ϕ22 =  −Z 0 X v . 0 −Z Y w From −Y X u 0 = −Z 0 v = Z 2 u − Y Z v + X Z w 0 −Z w

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we get Z u − Y v + X w = 0, which shows that the fourth column of ϕ22 is a linear combination of the first three columns. Thus ϕ is equivalent to a matrix having a zero column, contradiction. The case m = 2 is excluded by using Remark 6.3. We conclude that m = 3 and, from what was said above, that we have     X −Y X 0 ψ21 ∼  Y  , ϕ22 ∼  −Z 0 X . Z 0 −Z Y Thus far we have obtained the desired resolution of F in the cases n ≥ 6. The cases n = 4 and n = 5 are completely analogous. From our concrete description of ϕ22 we see that C ' O(1). Since F surjects onto G, the latter has support of dimension zero or one. Thus, at least one of the maximal minors of ϕ11 must be a nonzero polynomial. We can apply Remark 6.5 to conclude that Ker (ϕ11 )' O(−d + 1) for some integer d ≥ 3. We have t t +3−d (d −2)(d −3) t +1 − (n −2) + = (n −d)t + PG (t) = (n −3) . 2 2 2 2 The sheaf G violates the semistability of F precisely when (d − 2)(d − 3) 3 < , 2(n − d) n

that is, n(d − 5) < −6.

Thus, we cannot have d = 3 and, if d = 4, then n ≤ 6. We conclude that F is an extension 0 → OC (1) → F → G → 0 with deg(C) = d ≥ 4, respectively deg(C) ≥ 5 in the case n ≥ 7. Finally, we cannot have 0 ϕ 0 ϕ11 ∼ with ϕ 0 : m O(−2) → m O(−1), 1 ≤ m ≤ n − 3. ? ? Indeed, if this were the case, then, since ϕ11 has at least one nonzero maximal minor, we would get det(ϕ 0 ) 6= 0 and a surjection F → Coker (ϕ 0 ) onto a sheaf with Hilbert polynomial P(t) = mt. This would contradict the semistability of F. Lemma 6.7. Let C ⊂ P2 be a curve given by the equation f = 0, where f (X, Y, Z ) is a homogeneos polynomial. Let I ⊂ OC be a sheaf of ideals. Then there is a homogeneous polynomial g(X, Y, Z ) dividing f such that the sheaf of ideals J ⊂ OC generated by g satisfies: I ⊂ J and J/I is supported on finitely many points.


105

Proof. Dehomogenizing in a suitable open affine subset we reduce the problem to the following: let f (X, Y ) be a polynomial in k[X, Y ]. Let I ⊂ k[X, Y ] be an ideal containing f . Then there is a polynomial g(X, Y ) dividing f such that I ⊂ hgi and hgi/I is supported on finitely many points. Let f = f 1n 1 . . . f κn κ be the decomposition of f into irreducible factors. Let I = q1 ∩ . . . ∩ qm ∩ a1 ∩ . . . ∩ al be a primary decomposition of I . Here m ≤ κ, qi is a primary ideal associated to h f i i and a1 , . . . , al are primary ideals associated to maximal ideals m1 , . . . , ml . Put q = q1 ∩ . . . ∩ qm . We notice that q/I is supported on m1 , . . . , ml . For 1 ≤ i ≤ m let ri be the largest integer such that qi ⊂ h f iri i. We claim that g = f 1r1 · . . . · f mrm is the desired polynomial. To prove this it is enough to show that hgi/q is supported on finitely many points. Since localization commutes with intersections it is enough to show that each h f iri i/qi is supported on finitely many points. So far we have reduced the problem to the following: let f ∈ k[X, Y ] be an irreducible polynomial. Let q ⊂ k[X, Y ] be a primary ideal associated to h f i. Let r ≥ 1 be the largest integer such that q ⊂ h f r i. Then h f r i/q is supported on finitely many points. We may assume that q is not a power of h f i. Let s be the smallest integer such that h f s i ⊂ q. We will prove the above statement by induction on s. If s = r + 1 then h f r i/h f r +1 i h f r i/q ' q/h f r +1 i can be regarded as the structure sheaf of a proper subscheme of the scheme X ⊂ P2 given by { f = 0}. This is so because h f r i/h f r +1 i ' k[X, Y ]/h f i

as k[X, Y ]- modules.

But X is an irreducible scheme of dimension one, hence any proper subscheme has dimension zero. Assume now that s > r + 1 and the statement is true for any ideal q0 satisfying 0 q ⊂ h f r i, q0 * h f r +1 i, h f s−1 i $ q0 . Such an ideal is q0 = q + h f s−1 i. By the induction hypothesis we know that h f r i/q0 is supported on finitely many points. To finish the proof it is enough to show that q0 /q is supported on finitely many points. But q0 /q ' h f s−1 i/q ∩ h f s−1 i.

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If q ∩ h f s−1 i 6= h f s i then the right-hand side is supported on finitely many points by the first step in the induction argument. Now choose h ∈ q \ h f r +1 i. Then f s−r −1 h ∈ q ∩ h f s−1 i \ h f s i. This finishes the proof of the lemma. In the remaining part of this section we will seek more precise information about the morphisms occuring in Claim 6.6. For a start, assume that F is an arbitrary sheaf having a resolution as in Claim 6.6; we determine which subsheaves F0 ⊂ F are destabilizing. Let G0 be the image of F0 in G and let I(1) be the preimage of F0 in OC (1). Here I is the ideal sheaf of a subscheme of C. By Lemma 6.7 we can find a curve C 0 ⊂ C such that the ideal sheaf J of C 0 contains I and PI(1) (t) = PJ(1) (t) − c, where c is a nonnegative integer. From the exact sequence 0 → I(1) → F0 → G0 → 0 we get PF0 (t) = PI(1) (t) + PG0 (t) = PJ(1) (t) + PG0 (t) − c. Put κ = deg(C 0 ). We allow κ = 0 for the case J = OC . From the exact sequence 0 → O(−d + 1) → O(−κ + 1) → J(1) → 0 we see that h 0 (J(1)) = 0 if κ ≥ 2. But then h 0 (I(1)) = 0, forcing the map H 0 (F0 ) → H 0 (G0 ) to be injective. Since the map F → G is zero on global sections we see that h 0 (F0 ) = 0. It follows that F0 does not violate the semistability of F. In the case κ = 0 we have PF/F0 (t) = c+ PG/G0 (t), hence F0 violates the semistability of F if and only if α1 (G/G0 ) > 0 and α0 (G/G0 ) + c 3 < . α1 (G/G0 ) n Assume now that κ = 1. We have PF/F0 (t) = POC (1)/I(1) (t) + PG/G0 (t) = t + 2 + c + PG/G0 , hence F0 violates semistability if and only if 2 + c + α0 (G/G0 ) 3 < . 1 + α1 (G/G0 ) n Now the exact sequence 0 → OC (1) → F → G → 0 together with the hypothesis h 1 (F(i)) = 0 for i ≥ 0 give h 1 (G(i)) = 0 for i ≥ 0. This and the exact sequence 0 → G0 → G → G/G0 → 0


107

yield h 1 (G/G0 (i)) = 0 for i ≥ 0. In particular α0 (G/G0 ) =h 0 (G/G0 ) ≥ 0. This eliminates the case α1 (G/G0 ) = 0 from above. We summarize our findings so far: Remark 6.8. F is semistable if and only if there are no quotients sheaves E of G satisfying 3 α (E) 6= 0. n 1 One direction was proved in the discussion above. The other direction follows by taking κ = 0 and c = 0, in other words taking F0 to be the preimage of G0 , where G0 is the kernel of the surjection G → E. h 1 (E) = h 1 (E(1)) = 0

and

0 ≤ α0 (E)
m + 1 − r > 0, i.e., m + 1 > r > s − 1, or r = m + 1. In the first case ϕ11 ∼

ψ 0 ? ?

with ψ an r × s-matrix. Since at least one of the maximal minors of ϕ11 is nonzero we must have r = s. But then our assumption on ϕ11 is contradicted. Assume now that r = m + 1, so α is surjective. If β is not surjective we get the same contradiction as above. Finally, if β is surjective then ϕ11 ∼

ϕ0 0 . ? ?

Also, 1 = α0 (E) < (3/n)α1 (E) = 3m/n forces (n/3)+1 < m +1, so our assumption on ϕ11 is contradicted. Claim 6.11. Let W be the space of morphisms ϕ : (n − 2) O(−2) ⊕ 3O(−1) → (n − 3) O(−1) ⊕ 3O. Let 3 = (λ1 , λ2 , µ1 , µ2 ) be a polarization satisfying λ1 < µ1
1 − 8n 1 m 2 λ2 > 1 −

if m 1 ≤ 3, if m 1 > 3.

Taking m 1 = 2, m 2 = n − 2, n 1 = n these conditions become λ1
0, then ∂1 can be placed inside the circle of radius sin γ1 /|H |, and the inequality (3) cannot be satisfied. Hence, we must have c ≤ 0. Now we suppose that 6 is a spanner in a triangular pyramid. For completeness, we include the proof of balancing formulas for spanners in triangular pyramids; see [Concus et al. 2001]. For i = 1, 2, 3, let γi be the contact angles, let i be the wetted regions, and let Ni be the unit vectors perpendicular to the faces of

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SUNG-HO PARK

the pyramid that are compatible with the surface normal N of 6. By integrating 1X = 2H NE over 6 and using the divergence theorem, we have Z 6

It is clear that

2H NE d S =

Z 6

Z

1X d S =

Z nEds = ∂i

∂i

Z nEds = ∂6

3 Z X i=1

nEds. ∂i

(cos γi νi + sin γi Ni )ds,

where νi is the outward conormal of i along ∂i . Since γi is constant, Z sin γi Ni ds = sin γi |∂i |Ni . ∂i

R From the divergence theorem, we have ∂ j cos γi νi ds = 0 for each connected i j component ∂i of ∂i , and Z 3 Z X E NdS + Ni d S = 0. 6

i=1

i

Therefore we have 3 Z 3 3 3 X X X X 2H Ni d S + sin γi |∂i |Ni = 2H |i |Ni + sin γi |∂i |Ni = 0. i=1

i

i=1

i=1

i=1

Since N1 , N2 , and N3 are linearly independent (that is, a triangular pyramid is unbalanced), we obtain the balancing formulas for spanners in triangular pyramids: sin γi |∂i | + 2H |i | = 0

for i = 1, 2, 3.

The same argument can be applied to 1X = n H NE to produce the balancing formulas for n-dimensional spanners in unbalanced polyhedral cones in the Euclidean space En+1 : sin γi |∂i | + n H |i | = 0. Since the balancing formulas hold for spanners in triangular pyramids, we have Theorem 2. Let 6 be a compact embedded cmc surface in a triangular pyramid. Suppose that 6 does not meet the edge and the vertex of the pyramid, and suppose that the contact angles between 6 and the faces of the pyramid are constant on each face of the pyramid. For i = 1, 2, 3, let γi be the contact angles, and let i be the planar domains bounded by the boundary curves on each face (the wetted regions). Then (4)

|i | ≥

sin2 γi π H2

for i = 1, 2, 3.

ON A NECESSARY CONDITION FOR SPANNERS IN A WEDGE

165

From the balancing formulas for higher dimensional spanners together with the isoperimetric inequality for domains in En , we have Theorem 3. Let M be a compact embedded cmc hypersurface in an unbalanced polyhedral cone in En+1 bounded by (n−1)-dimensional submanifold(s) on each face of the cone. Suppose that M does not meet the edge and the vertex of the cone, and suppose that the contact angles between M and the faces of the cone are constant on each face of the cone. Let γi be the contact angles and i be the domains on each face of the cone bounded by the boundary submanifold(s) on each face (the wetted regions). Then we have sin γ n i (5) |i | ≥ ω(n), |H | where ω(n) is the volume of the unit ball in En . In unbalanced polyhedral cones, there exists a unique spherical spanner for given mean curvature H and contact angles γi ’s. The inequalities (4) and (5) say that the wetted regions of a spanner in unbalanced polyhedral cone have bigger or equal area than the wetted region of the spherical spanner. In [Park 2005], it was shown that every ring type spanner in a wedge is actually spherical. In this context, we ask two questions: •

Is every spanner in a wedge or in a triangular pyramid spherical?

•

Is every spanner in an unbalanced polyhedral cone in En+1 spherical? References

[Concus et al. 2001] P. Concus, R. Finn, and J. McCuan, “Liquid bridges, edge blobs, and Scherktype capillary surfaces”, Indiana Univ. Math. J. 50:1 (2001), 411–441. MR 2002g:76023 Zbl 0996. 76014 [McCuan 1997] J. McCuan, “Symmetry via spherical reflection and spanning drops in a wedge”, Pacific J. Math. 180:2 (1997), 291–323. MR 98m:53013 Zbl 0885.53009 [Park 2005] S.-h. Park, “Every ring type spanner in a wedge is spherical”, Math. Ann. 332:3 (2005), 475–482. MR 2006h:53008 Zbl 02190811 Received March 7, 2007. Revised July 28, 2007. S UNG - HO PARK KOREA I NSTITUTE FOR A DVANCED S TUDY H OEGIRO 87 207-43 C HEONGNYANGNI 2- DONG , D ONGDAEMUN - GU S EOUL 130-722 KOREA [email protected]


MODULE SUPERSINGULIER, FORMULE DE GROSS–KUDLA ET POINTS RATIONNELS DE COURBES MODULAIRES M ARUSIA R EBOLLEDO We show how the Gross–Kudla formula about triple product L-functions allows us to construct degree-zero elements of the supersingular module annihilated by the winding ideal. Using the method of Parent, we apply those results to the study of rational points on modular curves, determining a set of primes of analytic density 1−9/210 for which the quotient of X 0 ( p r ) (r > 1) by the Atkin–Lehner operator w p r has no rational points other than the cusps and the CM points.

Introduction Pour N > 0 un entier, notons X 0 (N ) la courbe modulaire sur Q classifiant grossièrement les courbes elliptiques généralisées munies d’une N -isogénie. La motivation initiale de ces travaux est l’étude des points rationnels du quotient X 0+ ( pr ) de X 0 ( pr ) par l’involution d’Atkin–Lehner w pr pour p un nombre premier et r > 1 un entier. Pour cela, nous reprenons une méthode de Parent [2005] s’inspirant des travaux de Momose [1984; 1986; 1987] et faisant appel au module supersingulier. Fixons p > 3 un nombre premier et F¯ p une clôture algébrique de Fp . Nous appelons module supersingulier le Z-module libre P engendré par l’ensemble fini S = {x 0 , . . . , x g } des classes d’isomorphismes de courbes elliptiques supersingulières sur F¯ p . Notons P0 le sous-groupe de P constitué des e´ léments de degré nul. On peut munir P0 d’une action de l’anneau T engendré par les opérateurs de Hecke agissant sur l’espace vectoriel S2 (00 ( p)) des formes paraboliques de poids 2 pour 00 ( p) (voir 1A). Notons P0 [Ie ] l’ensemble des e´ léments de P0 annulés par l’idéal d’enroulement Ie ⊂ T c’est-à-dire l’annulateur des formes primitives f ∈ S2 (00 ( p)) telles que L( f, 1) 6= 0. Pour j ∈ P1 (F¯ p ) un invariant non supersingulier, considérons l’application (1)

ιj :

P → F¯ p

Pg

i=0 λi x i

7→

Pg

i=0

λi j − ji

MSC2000: primary 14G05, 11G05, 11G18; secondary 14G10, 11R52. Mots-clefs: rational points on modular curves, supersingular module, special values of L-functions. 167

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où, pour i ∈ {0, . . . , g}, ji est l’invariant d’une courbe elliptique E i ∈ xi . Parent a mis en e´ vidence le critère suivant (voir propositions 3.1 et 3.2 de [Parent 2005]) : (C) Soit p ≥ 11. Supposons que pour tout j ∈ Fp non supersingulier, il existe x ∈ P0 [Ie ] tel que ι j (x) 6= 0. Alors X 0+ ( pr )(Q) est trivial c’est-àdire ne contient que des pointes et des points CM. Pour A un anneau et f : M → N un homomorphisme de Z-modules, on note M A = M ⊗ A et on note encore f : M A → N A l’homomorphisme de A-modules obtenu par extension des scalaires. Soient n le numérateur de ( p − 1)/12 et π 0 : PZ[1/n] → P0Z[1/n] la projection orthogonale (voir Section 1C). Pour D > 0 notons γ D ∈ PQ le D-ième e´ lément de Gross (voir 3A). Notons Disc( p) l’ensemble des discriminants quadratiques imaginaires1 premiers a` p. Parent déduit de la formule de Gross–Zhang que γ D0 := π 0 (γ D ) ∈ P0 [Ie ]Q pour −D ∈ Disc( p). Considérons une autre famille d’éléments du module supersingulier : (2)

ym =

g D X

Tm xi ,

i=0

E xi x ∈ P et wi i

ym0 = π 0 (ym ) ∈ P0Z[1/n]

(m ≥ 1)

où h , i est l’accouplement bilinéaire sur P défini en (3) et wi = |Aut(E i )|/2 pour E i ∈ xi . Théorème 0.1. Pour tout entier m ≥ 1, ym0 ∈ P0 [Ie ]Q . Pg Notons a E = i=0 xi /wi l’élément d’Eisenstein (voir le Section 1C). L’assertion précédente peut se déduire de la formule de Gross–Kudla ou, pour certaines valeurs de m, de la formule de Gross–Zhang et de la proposition suivante Proposition 0.2. On a ym = (m) a E +

X

γd

(m ≥ 1)

(s,d)∈Z2 4m−s 2 =dr 2 >0

où (m) = 1 si m est un carré et (m) = 0 sinon. La démonstration de la proposition 0.2 s’inspire du calcul classique qui permet d’établir la formule des traces d’Eichler (voir [Eichler 1955], [Gross 1987] et la Section 3B). Parent [2005] montre que {γ D0 , −D ∈ Disc( p)} engendre le Q-espace vectoriel P0 [Ie ]Q . Nous montrons au Section 3B les propositions suivantes : Proposition 0.3. Le Q-espace vectoriel engendré par (ym0 )m≥1 est égal au Qespace vectoriel engendré par (ym0 )1≤m≤g+1 . 1 On appelle ici discriminant quadratique imaginaire le discriminant d’un ordre d’un corps qua-

dratique imaginaire. C’est donc un carré mutiplié par un discriminant fondamental.

MODULE SUPERSINGULIER, FORMULE DE GROSS–KUDLA ET POINTS RATIONNELS 169

Proposition 0.4. Le TQ -module engendré par {ym0 , m ≥ 1} est égal à P0 [Ie ]Q . Pour√ −d < 0 un discriminant fondamental, notons εd le caractère non trivial de Gal(Q( −d)/Q). Les propositions 0.2, 0.3 et 0.4 entraˆınent une version précise d’un théorème de non-annulation : Corollaire 0.5. Si f est une forme primitive de poids 2 pour 00 ( p) telle que L( f, 1) 6= 0, alors il existe d ≤ 4g + 4 tel que L( f ⊗ εd , 1) 6= 0. Pour (m, p) = 1, ym e´ numère les boucles du graphe des m-isogénies e´ tudié par Mestre et Oesterlé [≥ 2008]. Cela permet de faire les calculs (Section 4B) conduisant au Théorème 0.6 suivant. Considérons le nombre premier p0 = 45321935159. Soit C l’ensemble des nombres premiers p qui sont un carré modulo 3, 4 et 7 et tels que l’une des conditions suivantes soit vérifiée : (1) p carré modulo 5, 11, 19, 23, 43, 67, 163, non carré modulo 8 ; (2) p carré modulo 8, 11, 19, et modulo au moins deux des nombres premiers 43, 67, 163, et vérifiant l’une des conditions suivantes (a) p carré modulo 5 ; (b) p non carré modulo 5 et 23 ; (c) p non carré modulo 5 et carré modulo 23, 59, 71 ; (d) p non carré modulo 5, 59, 71 et carré modulo 23 ; (3) p carré modulo 5, 8, 11, 43, 67, 163, non carré modulo 19 et l’une des conditions suivantes est vérifiée : (a) p carré modulo 23 ; p p p = 1. 31 36319 p0 (4) p carré modulo 5, 8, 19, 43, 67, 163, non carré modulo 11 et p carré modulo au moins un des nombres : 23, 797. (b) p non carré modulo 23 et

Théorème 0.6. Si p ≥ 11, p 6= 13 et p 6∈ C, alors X 0+ ( pr )(Q) est trivial. L’ensemble C est de densité analytique 9/210 . Parent [2005] avait obtenu un résultat analogue avec une densité de 7/29 . Le cas r = 2 de ce théorème constitue une avancée en direction du cas normalisateur d’un Cartan déployé d’un problème de Serre sur la torsion des courbes elliptiques. (Pour un e´ noncé de ce problème, se reporter a` [Serre 1972; Serre 1986, p. 288].) La liste des nombres premiers 11 ≤ p ≤ 50000, p 6= 13 dans C est donnée dans la Table 3 (page 182).

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1. Préliminaires sur le module supersingulier 1A. Réalisations géométriques et opérateurs de Hecke. Soit e T l’anneau engendré par l’action des opérateurs de Hecke Tm , m ≥ 1 sur le C-espace vectoriel M2 (00 ( p)) des formes modulaires de poids 2 pour 00 ( p). Cette action se factorise par T sur S2 (00 ( p)). Soient X 0 ( p)Z la normalisation de P1Z dans X 0 ( p) via le morphisme composé X 0 ( p) → X 0 (1) ∼ = P1Q . La fibre X 0 ( p)Fp de X 0 ( p)Z en p est constituée 1 de deux copies de PFp qui sont e´ changées par l’opérateur d’Atkin–Lehner w p et se coupent transversalement. Les points doubles de X 0 ( p)Fp sont en correspondance bijective avec les classes x0 , . . . , x g de S et g n’est autre que le genre de X 0 ( p). Notons J0 ( p) la jacobienne de X 0 ( p) et Jela jacobienne généralisée de X 0 ( p) relativement aux pointes. L’anneau e T (resp. T) est isomorphe a` l’anneau engendré par e les endomorphismes de J (resp. J0 ( p)) provenant des correspondances de Hecke sur X 0 ( p). Soient J0 ( p)Z et JeZ les modèles de Néron respectifs de J0 ( p) et Je sur Z. Le groupe P = Z[S] (resp. P0 ) s’identifie au groupe des caractères de la composante neutre de la fibre en p de JeZ (resp. J0 ( p)Z ) (voir [Raynaud 1991] et [de Shalit 1995, 2.3]). Cela définit par transport de structure une action de e T sur P qui laisse stable P0 et se factorise par T sur P0 . L’action de e T sur la classe d’isomorphisme [E] d’une courbe elliptique E supersingulière sur F¯ p est donnée P par Tm ([E]) = C [E/C] (m ≥ 1), où C parcourt l’ensemble des sous-schémas en groupes finis d’ordre m de E (voir [Raynaud 1991] ou [Mestre et Oesterlé ≥ 2008, 1.2.1]). Sur la base (xi )0≤i≤g de P, l’action de Tm (m ≥ 1) est donnée par la transposée de la matrice d’Eichler–Brandt B(m) = (Bi, j (m))0≤i, j≤g (voir [Gross 1987, sections 1 et 4]). 1B. Accouplement bilinéaire. Soit δ le dénominateur de ( p − 1)/12. Rappelons que wi = |Aut(E i )|/2 (i ∈ {0, . . . , g}, E i ∈ xi ) vérifient g Q

wi = δ

i=0

et

g P

1/wi = ( p − 1)/12.

i=0

Le Z-module P est muni de l’accouplement bilinéaire non dégénéré h , i : P × P → Z défini par (3)

hxi , x j i = wi δi, j

(0 ≤ i, j ≤ g)

où δi, j est le symbole de Krönecker. Les opérateurs de Hecke sont auto-adjoints pour h , i. Cet accouplement induit un homomorphisme injectif de e T-modules de ˇ e P dans P = Hom(P, Z) (sur lequel T agit par dualité) de conoyau isomorphe a` ˇ au sous-e Z/δ Z identifiant P T-module g L i=0

Z

xi wi


de PQ (voir [Gross 1987] ou [Emerton 2002, lemme 3.16]). L’accouplement caˇ → Z e´ tend donc l’accouplement h , i et sera encore noté h , i. nonique P × P L’homomorphisme injectif de T-modules de P0 dans Pˇ0 = Hom(P0 , Z) induit par h , i est de conoyau isomorphe a` Z/n Z (loc. cit.). Sur Pˇ0 /P0 qui s’identifie au groupe des composantes de la fibre en p de J0 ( p)Z , l’accouplement h , i n’est autre que l’accouplement de monodromie (voir [Illusie 1991] ou l’appendice de [Bertolini et Darmon 1997]). P Notons m≥0 am ( f )q m le développement de Fourier a` l’infini de f ∈ M2 (00 ( p)). Considérons le Z-module M des formes modulaires f telles que a0 ( f ) ∈ Q et am ( f ) ∈ Z pour m ≥ 1 et M0 le sous-Z-module de M constitué des formes paraT sur M2 (00 ( p)) laisse stables M et M0 et se factorise par boliques2 . L’action de e 0 T T sur M . On a MC = M2 (00 ( p)) et M0C = S2 (00 ( p)). L’accouplement sur M × e e défini par ( f, T ) 7→ a1 ( f | T ) induit un isomorphisme de T-modules (resp. de T-modules) ∼ ∼ M− → Hom(e T, Z) (resp. M0 − → Hom(T, Z)) (voir [Ribet 1983, théorème 2.2 ; Emerton 2002, proposition 1.3]). Les homomorphismes de e T-modules et T-modules ∼ ˇ e θ : P ⊗e T P → Hom(T, Z) = M P 1 m x ⊗e T y 7 → 2 (deg x. deg y) + m≥1 hTm x, yi q et θ 0 : P0 ⊗T Pˇ0 → Hom(T, Z) ∼ = M0 qui se déduisent de l’accouplement h , i, sont des surjections (voir [Emerton 2002, théorème 0.10]). e -module P. Les e 1C. Le T TQ -modules PQ et MQ et les TQ -modules P0Q et M0Q sont libres de rang 1 (voir par exemple [Gross 1987] et [Miyake 1989]). Appelons forme de Hecke une forme modulaire de poids 2 pour 00 ( p) propre pour tous les opérateurs de Hecke et normalisée, et forme primitive une forme de Hecke parabolique. On note Prim l’ensemble des formes primitives. Les idempotents primitifs de e TQ¯ sont en correspondance bijective avec les formes de Hecke et engendrent les sous-e TQ¯ -modules irréductibles de e TQ¯ . On note 1 f l’idempotent primitif associé a` une forme de Hecke f . Les opérateurs de Hecke e´ tant autoadjoints pour h , i, le e TQ¯ -module PQ¯ se décompose en somme directe de sous-espaces propres orthogof ¯ -droite naux. Pour f une forme primitive, le sous-espace PQ¯ = 1 f PQ¯ est une Q 0 dont on choisit un vecteur directeur a f . Notons σ (m) la somme des diviseurs de 2 Pour f ∈ M, on a en fait δ a ( f ) ∈ Z (voir [Emerton 2002, proposition 1.1]). Attention, les 0

notations diffèrent de celles adoptées dans [Emerton 2002] où M est noté N.

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m premiers a` p. Le sous-espace propre associé a` la série d’Eisenstein normalisée P E = ( p − 1)/24 + m≥1 σ 0 (m)q m est engendré par l’élément d’Eisenstein g X xi ˇ ⊂ PZ[1/δ] . aE = ∈P wi i=0

De plus, a E vérifie hx, a E i = deg x (x ∈ P). Par conséquent, P E := Z.a E est le ˇ → Z. On note Z-module orthogonal a` P0 pour h , i : P × P π 0 : PZ[1/n] → P0Z[1/n] 12 deg(x) a E x 7→ x − p−1 la projection orthogonale. e¯ 1D. Produits tensoriels. Considérons le e T-module P ⊗e T P. Les sous-espaces TQ g g ecrivant l’enTQ¯ -modules PQ¯ ⊗e propres de PQ¯ ⊗e ¯ sont les e TQ¯ PQ TQ¯ PQ ¯ pour g d´ semble des formes de Hecke et on a les décompositions en sous-espaces deux a` deux orthogonaux M g g E E 0 0 0 0 PQ ⊗e PQ¯ ⊗TQ¯ PQ¯ . ¯ ⊗TQ¯ PQ ¯ = TQ PQ )⊕(PQ ⊗TQ PQ ) et PQ TQ PQ = (PQ ⊗e g∈Prim g 1 f PQ¯

h En effet, pour toutes formes de Hecke f, g et h, on a ⊗e TQ¯ PQ ¯ 6 = 0 si et P g g h h seulement si f = g = h. Donc si g 6= h, on a PQ¯ ⊗e TQ¯ PQ TQ¯ PQ f 1 f PQ ¯ = ¯ = 0. ¯ ⊗e ⊗3

Considérons a` présent les e T⊗3 -modules P⊗3 , M⊗3 , et les T⊗3 -modules P0 ⊗3 et M0 . Par fonctorialité des algèbres tensorielles, on déduit de l’accouplement ˇ ⊗3 → Z. De même, le produit scalaire h , i un accouplement3 h , i⊗3 : P⊗3 × P 0 de Petersson ( , ) sur MC × MC (normalisé comme dans [Gross 1987] (7.1)) définit ⊗3

le produit scalaire de Petersson ( , )⊗3 sur M0C × MC ⊗3 (normalisé par [Gross et Kudla 1992] (11.3)). ⊗3 ⊗3 ⊗3 ⊗3 0 ⊗3 Les e TQ -modules M⊗3 et P0Q sont libres Q et PQ et les TQ -modules MQ de rang 1. Les opérateurs de Hecke triples de e T⊗3 sont autoadjoints pour ( , )⊗3 ⊗3 et h , i⊗3 . Les idempotents primitifs de e TQ¯ sont de la forme 1 F = 1 f 1 ⊗ 1 f2 ⊗ 1 f3 pour F = f 1 ⊗ f 2 ⊗ f 3 parcourant l’ensemble des formes de Hecke triples (c’est¯ -espace vectoriel a` -dire telles que f 1 , f 2 et f 3 soient des formes de Hecke). Le Q ⊗3 F ¯ PQ¯ = 1 F PQ¯ est une Q-droite de vecteur directeur A F = a f 1 ⊗Q¯ a f2 ⊗Q¯ a f3 . On a les décompositions en sous-espaces propres deux a` deux orthogonaux ⊗3 M M F F P⊗3 = P et P0Q¯ = PQ ¯ ¯, ¯ Q Q F 3 donné par ha ⊗ a ⊗ a , b ⊗ b ⊗ b i⊗3 = Q3 ha , b i 1 2 3 1 2 3 i=1 i i

F


la somme directe portant respectivement sur l’ensemble des formes de Hecke triples et l’ensemble des formes primitives triples. On vérifie aisément que pour toute forme de Hecke triple F, on a 1F X =

(4)

hA F , X i⊗3 AF hA F , A F i⊗3

⊗3 (X ∈ PQ ¯ ).

2. Formule de Gross–Kudla et éléments de P0 [Ie ] Soit

g X 1 ⊗3 x ∈ P⊗3 13 = Q wi i i=0

l’élément diagonal de Gross–Kudla. Posons s : P ⊗ P P ⊗e T P la surjection canonique, ¯ 3 = (1 ⊗Q s)(13 ) ∈ PQ ⊗Q PQ ⊗e 1 TQ PQ et ¯ 03 = (π 0 ⊗Q s)(13 ) = (π 0 ⊗Q 1)(1 ¯ 3 ) ∈ P0Q ⊗Q PQ ⊗e 1 TQ PQ . Gross et Kudla [1992, théorème 11.1] ont montré que pour F une forme primitive triple, on a (F, F)⊗3 L(F, 2) = h1 F 13 , 1 F 13 i⊗3 . 4π p En particulier, lorsque F = f ⊗ h ⊗ h ( f, h ∈ Prim), par [Gross et Kudla 1992, (11.7)], on obtient (5)

L( f, 1)L( f ⊗ Sym2 h, 2) =

(F, F)⊗3 h1 F 13 , 1 F 13 i⊗3 . 4π p

Nous allons voir que cela entraˆıne le Théorème 2.1. On a ¯ 03 1

12 ¯3− =1 a E ⊗Q p−1

et

X g

xi ⊗e TQ

i=0

xi wi

¯ 03 ∈ P0 [Ie ]Q ⊗ (P0Q ⊗TQ P0Q ). 1 Remarque 2.2. Le T-module P0 est localement libre après extension des scalaires a` Z[1/2] (voir l’introduction de [Emerton 2002]), par conséquent il y a au pire de la 2-torsion dans P0 [Ie ] ⊗ (P0 ⊗T P0 ). Ainsi, puisque δ13 ∈ P⊗3 , on a même ¯ 0 ∈ P0 [Ie ] ⊗ (P0 ⊗T P0 ) a` un e´ lément de 2-torsion près. nδ 1 3 Démonstration. On a ¯ 03 = 1

g X i=0

xi −

xi 12 a E ⊗Q xi ⊗e , TQ p−1 wi

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MARUSIA REBOLLEDO

ce qui prouve la première assertion du Théorème 2.1. On a dans P0Q¯ ⊗Q¯ (PQ¯ ⊗e ¯) TQ¯ PQ X X ¯ 03 = ¯ 3) + ¯ 3) 1 (1 f ⊗ 1h )(1 (1 f ⊗ 1 E )(1 f,h∈Prim

f ∈Prim

X X (1 f ⊗ 1h ⊗ 1h )(13 ) + (1 f ⊗ 1 E ⊗ 1 E )(13 ) . = (1 ⊗ s) f,h∈Prim

f ∈Prim

Pour tout f ∈ Prim, on a (voir (4)) h13 , a f ⊗ a E ⊗ a E i⊗3 (a f ⊗ a E ⊗ a E ). ha f ⊗ a E ⊗ a E , a f ⊗ a E ⊗ a E i⊗3 Pg Or h13 , a f ⊗ a E ⊗ a E i⊗3 = i=0 w1i hxi , a f i hxi , a E i2 = ha E , a f i = 0. Par conséquent, X 0 ¯ (1 f ⊗ 1h ⊗ 1h )(13 ) 13 = (1 ⊗ s) (1 f ⊗ 1 E ⊗ 1 E )(13 ) =

f,h∈Prim

X X = (1 ⊗ s) (1 f ⊗ 1h ⊗ 1h )(13 ) + (1 f ⊗ 1h ⊗ 1h )(13 ) . f,h∈Prim L( f,1)6 =0

f,h∈Prim L( f,1)=0

Puisque ( , )⊗3 et h , i⊗3 sont définis positifs, on déduit de (5) que lorsque f ∈ Prim est telle que L( f, 1) = 0, alors (1 f ⊗ 1h ⊗ 1h )(13 ) = 0 (h ∈ Prim). Par ailleurs, rappelons que l’idéal Ie de T est l’annulateur de l’ensemble des formes paraboliques f pour lesquelles L( f, 1) 6= 0 (voir [Merel 1996]). On en déduit que, lorsque f ∈ Prim est telle que L( f, 1) 6= 0, le vecteur propre associé a f est dans P0Q¯ [Ie ] et donc (1 f ⊗ 1h ⊗ 1h )(13 ) ∈ P0 [Ie ]Q ⊗Q P0Q ⊗Q P0Q (h ∈ Prim). On déduit immédiatement du Théorème 2.1 le corollaire suivant : ¯ 0) ∈ Corollaire 2.3. Pour toute forme Q-linéaire φ sur P0Q ⊗TQ P0Q , on a (1⊗Q φ)(1 3 P0 [Ie ]Q . Par définition de θ, on a (6)

¯ 3 ) et ym = (1 ⊗ (am ◦ θ ))(1

¯ 03 ). ym0 = (1 ⊗Q (am ◦ θ 0 ))(1

Le Théorème 0.1 e´ noncé dans l’introduction se déduit alors du corollaire 2.3 appliqué a` φ = am ◦ θ 0 . Remarque 2.4. Toute forme linéaire sur P0Q ⊗TQ P0Q s’obtient comme combinaison linéaire des formes linéaires am ◦ θ 0 (m ≥ 1) car θ 0 : PQ ⊗TQ P0Q → M0Q est un isomorphisme. Il n’est donc pas restrictif de considérer les combinaisons linéaires d’éléments ym0 comme nous le faisons par la suite.


3. Les éléments ym 3A. Comparaison avec les éléments de Gross. Soit D > 0. Notons O−D l’ordre quadratique de discriminant −D s’il existe et u(−D) l’ordre de O∗−D /h±1i. Pour i ∈ {0, . . . , g}, les anneaux Ri = End E i , i ∈ {0, . . . , g}, sont des ordres maximaux de l’algèbre de quaternions B sur Q ramifiée en p et ∞ ; on note h i (−D) le nombre de plongements optimaux de O−D dans Ri modulo conjugaison par Ri∗ . Le D-ième e´ lément de Gross4 est défini par g

(7)

X 1 γD = h i (−D)xi ∈ PZ[1/6] . 2u(−D) i=0

On a γ D = 0 si −D n’est √ pas un discriminant quadratique imaginaire ou bien si p est décomposé dans Q( −D) (en effet, dans ce dernier cas, O−D ne se plonge pas dans l’algèbre de quaternions ramifiée en p et l’infini). Démonstration de la proposition 0.2. Soit m ≥ 1 un entier. L’opérateur Tm agissant Pg sur xi par la transposée de la matrice de Brandt B(m), on a ym = i=0 Bi,i (m)xi . Notons N la norme réduite et tr la trace sur B. On a 1 1 X Bi,i (m) = Card{b ∈ Ri ; N (b) = m} = Card(Ai (s, m)) 2wi 2wi s∈Z s 2 ≤4m

où Ai (s, m) = {b ∈ Ri ; N (b) = m, tr(b) = s}. Posons D = 4m − s 2 . Lorsque D = 0, ce qui est possible si et seulement si m est un carré, Ai (s, m) n’a qu’un seul e´ lément. Considérons maintenant le cas où D > 0. Les e´ léments de Ai (s, m) sont en bijection avec les plongements de O−D dans Ri . Pour chaque tel plongement f il existe un unique ordre O−d contenant O−D (D = dr 2 pour un certain r ) tel que f s’étende en un plongement optimal de O−d dans Ri . On a donc une partition G Ai (s, m) = Ai (s, m)d d∈N;∃r ∈N; dr 2 =D

où les e´ léments de Ai (s, m)d ⊂ Ai (s, m) correspondent aux plongements de O−D dans Ri qui s’étendent en un plongement optimal de O−d dans Ri . Comme Card(Ai (s, m)d ) = h i (−d) | Ri× /O× −d |= wi h i (−d)/u(−d), 4 Cet élément, introduit par Gross, est noté e

D dans [Gross 1987] et [Parent 2005].

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MARUSIA REBOLLEDO

X h i (−d) on a finalement Card(Ai (s, m)) = wi . Par conséquent, pour tout u(−d) entier m > 0, on a d∈N;∃r ∈N dr 2 =D

ym =

X

g X

X

(s,d)∈Z×N ∃r >0; 4m−s 2 =dr 2 >0 g X X h i (−d) aE + xi . 2u(−d) i=0 (s,d)∈Z×N 4m−s 2 =dr 2 >0

4m=s 2 i=0

= (m)

xi + 2wi

g X i=0

h i (−d) xi 2u(−d)

Remarque 3.1. Le raisonnement précédent est celui qui donne la formule d’Eichler pour la trace de Tm (voir [Eichler 1955] ou [Gross 1987]). On retrouve cette formule en identifiant les degrés de chacun des membres de l’égalité de la proposition 0.2. Remarque 3.2. La formule de Gross [1987, corollaire 11.6] montre que pour tout entier d < 0 premier a` p, l’élément γd0 est dans P0 [Ie ]Q (voir par exemple [Parent 2005]). La proposition 0.2 donne alors une nouvelle preuve du Théorème 0.1 dans le cas particulier où m est tel que tout entier d > 0 tel que 4m − s 2 = dr 2 soit premier a` p. ` titre d’exemple, voici la décomposition de ym comme combinaison linéaire A d’éléments de Gross pour m ≤ 13 : y1 = a E + 2γ3 + γ4 , y2 = 2γ4 + 2γ7 + γ8 , y3 = 3γ3 + 2γ8 + 2γ11 + γ12 , y4 = a E + 2γ3 + γ4 + 2γ7 + 2γ12 + 2γ15 + γ16 , y5 = 4γ4 + 2γ11 + 2γ16 + 2γ19 + γ20 , y6 = 2γ8 + 2γ15 + 2γ20 + 2γ23 + γ24 , y7 = 6γ3 + γ7 + 2γ12 + 2γ19 + 2γ24 + 2γ27 + γ28 , y8 = 2γ4 + 4γ7 + γ8 + 2γ16 + 2γ23 + 2γ28 + 2γ31 + γ32 , y9 = a E + 2γ3 + γ4 + 2γ8 + 2γ11 + 2γ20 + 2γ27 + 2γ32 + 2γ35 + γ36 , y10 = 4γ4 + 2γ15 + 2γ24 + 2γ31 + 2γ36 + 2γ39 + γ40 , y11 = 2γ7 + 2γ8 + γ11 + 2γ19 + 2γ28 + 2γ35 + 2γ40 + 2γ43 + γ44 , y12 = 3γ3 + 2γ8 + 2γ11 + 3γ12 + 2γ23 + 2γ32 + 2γ39 + 2γ44 + 2γ47 + γ48 , y13 = 6γ3 + 4γ4 + 2γ12 + 2γ16 + 2γ27 + 2γ36 + 2γ43 + 2γ48 + 2γ51 + γ52 . 3B. Espace vectoriel et module de Hecke engendrés par ces éléments. Nous démontrons dans ce paragraphe les propositions 0.3 et 0.4 ainsi que le corollaire


0.5. Faisons tout d’abord quelques observations. A une forme linéaire φ sur PQ , ¯ 0 ) ∈ M0 . Remarquons que gφ a associons la forme modulaire gφ = (φ ⊗Q θ 0 )(1 3 Q pour q-développement X (8) φ(ym0 )q m . m≥1

De plus, puisque φ ⊗Q¯ θ 0 est un homomorphisme de TQ¯ ⊗Q¯ TQ¯ -modules, on a X 0 ¯ (9) 1h gφ = (φ ⊗Q¯ θ ) (1g ⊗Q¯ 1h )(13 ) (h ∈ Prim). g∈Prim

Démonstration de la proposition 0.3. Il suffit de montrer que toute forme linéaire 0 est sur l’espace vectoriel engendré par (ym0 )m≥1 et qui s’annule en y10 , . . . , yg+1 nulle. Soit φ une telle forme linéaire. La forme différentielle ωφ de X 0 ( p) associée a` gφ a pour q-développement X dq φ(ym0 )q m . q m≥1

0 Si φ(y10 ) = 0, . . . , φ(yg+1 ) = 0, la forme différentielle holomorphe ωφ a un zéro d’ordre g en l’infini. L’infini n’étant pas un point de Weierstrass de X 0 ( p) (voir [Ogg 1978]), on en déduit que ωφ est nulle, d’où la proposition.

Pour démontrer la proposition 0.4 on a encore besoin de deux lemmes : Lemme 3.3. Soient f et h deux formes primitives. Les assertions suivantes sont équivalentes : a) L( f, 1)L( f ⊗ Sym2 h, 2) = 0 ;

b) 1h .gh .,a f i = 0.

Démonstration. D’après (9), on a 1h gh . ,a f i = h . , a f i ⊗Q¯ θ

0

X

¯ 3) . (1g ⊗Q¯ 1h )(1

g∈Prim

Puisque hag , a f i = 0 si g 6= f , on obtient ¯ 3 ). 1h gh.,a f i = (h . , a f i ⊗Q¯ θ 0 )(1 f ⊗Q¯ 1h )(1 Or l’application f

h . , a f i ⊗Q¯ θ 0 : PQ¯ ⊗Q¯ (P0Q¯ ⊗TQ¯ P0Q¯ ) → M0Q¯ est injective car θ 0 est un isomorphisme de TQ -modules libres (voir fin du Section ¯ 3 ) = 0 i.e. 1A). Par conséquent 1h gh . ,a f i = 0 si et seulement si (1 f ⊗Q¯ 1h )(1 si et seulement si (1 f ⊗Q¯ 1h ⊗Q¯ 1h )(13 ) = 0. D’après (5), ceci est e´ quivalent a` l’assertion a).

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MARUSIA REBOLLEDO

Lemme 3.4. Les conditions suivantes sont équivalentes : i) le TQ -module Y engendré par {ym0 , m ≥ 1} est égal à P0 [Ie ]Q ; ii) pour toute forme primitive f de poids 2 pour 00 ( p) telle que L( f, 1) 6= 0, il existe une forme primitive h de poids 2 pour 00 ( p) telle que L( f ⊗ Sym2 h, 2) 6= 0. ¯ -droites, on a Y = Démonstration. Les espaces TQ¯ -propres de P0Q¯ e´ tant des Q P0 [Ie ]Q si et seulement si pour tout f ∈ Prim tel que L( f, 1) 6 = 0, il existe m ≥ 1 tel que 1 f ym 6= 0. Par ailleurs, d’après le lemme 3.3, la condition ii) du lemme est vérifiée si et seulement si pour tout f ∈ Prim tel que L( f, 1) 6= 0, on a gh . ,a f i 6= 0 (car il existe alors h telle que 1h gh . ,a f i 6= 0). Puisque gh . ,a f i a pour q-développement P m eduit l’équivalence des assertions i) et ii). m≥1 hym , a f i q (voir (8)), on en d´ Démonstration de la proposition 0.4. D’après les travaux de Böcherer et SchulzePillot [1999] améliorés par Arakawa et Böcherer [2003, théorème 5.3], la condition ii) du lemme ci-dessus est toujours satisfaite. Démonstration du corollaire 0.5. Soit f ∈ Prim telle que L( f, 1) 6= 0. D’après les propositions 0.3 et 0.4, il existe alors 1 ≤ m ≤ g + 1 tel que 1 f ym 6= 0. Par conséquent, il existe D ≤ 4g +4 tel que 1 f γ√D 6= 0. On a (D, p) = 1 car D ≤ 4g +4. Soit −d le discriminant fondamental de Q( −D). En vertu des relations de norme [Bertolini et Darmon 1996] 2.4, on a γ D ∈ e T.γd . On en déduit que 1 f γd 6 = 0 et L( f ⊗ εd , 1) 6= 0 d’après la formule de Gross [1987, corollaire 11.6] généralisée par Zhang [2001, théorème 1.3.2]. 4. Points rationnels de courbes modulaires 4A. Une question théorique. On suppose désormais p ≥ 11. Soit Z( p) le localisé de Z en p. Soit j ∈ P1 (F¯ p ) non supersingulier. On e´ tend ι j a` PZ( p) en posant ι j (1/a) = a −1 . Supposons vérifiée la condition : (H)

Pour tout j ∈ Fp ordinaire, il existe m ≥ 1 tel que ι j (ym0 ) 6= 0.

D’après le Théorème 0.1 et le critère (C) e´ noncé dans l’introduction, on a alors p assez grand (par exemple p > 37), l’hypothèse (H) est vérifiée. Dans l’espoir d’obtenir un résultat sans les contraintes de congruence du Théorème 0.6, remarquons que l’on peut reformuler 0 ¯ 0 ) ∈ P0 cette condition. On a (1⊗θ 0 )(1 erons la forme parabolique 3 Z[1/n] ⊗M . Consid´ 0 a` coefficients dans F¯ p (i.e. l’élément de F¯ p ⊗ M ) définie par X 0+ ( pr )(Q) trivial. Actuellement, on ne sait pas si pour

¯ 03 ) ∈ M0¯ . g j = (ι j ⊗ θ 0 )(1 Fp P Cette forme modulaire a pour q-développement m≥1 ι j (ym0 ) q m . L’hypothèse (H) est donc e´ quivalente a` : (10)


(H0 )

Pour tout j ∈ Fp ordinaire, on a g j 6= 0.

4B. Calculs pratiques : démonstration du Théorème 0.6. Soit j ∈ Fp non supersingulier. On a ι j (ym0 ) =

g g X X 1 Bi,i (m) + 12 tr(B(m)) ∈ F¯ p . j − ji wi ( j − ji ) i=0

i=0

En effet, ym0

12 deg(ym )a E = ym − p−1

et ι j (a E ) =

g X i=0

1 . wi ( j − ji )

Le membre de droite de cette dernière e´ galité n’étant pas facile a` calculer, nous introduisons yk,m = tr B(m)yk − tr B(k)ym ∈ P0 [Ie ]Q

(0 < k < m)

et calculons ι j (yk,m ) pour k, m dans {2, 3, 5, 6, 7}. Lorsque (m, p) = 1, l’entier Bi,i (m) est la multiplicité de ji comme racine du polynôme modulaire φm (X, X ) dans F¯ p (voir [Igusa 1959] ou [Lang 1987, 5.3 théorème 5]). Les polynômes modulaires φm (X, X ) pour m ∈ {2, 3, 5, 6, 7}, donnés par Magma, sont : φ2 (X, X ) = −(X − 1 728)(X + 3 375)2 (X − 8 000), φ3 (X, X ) = −X (X − 54 000)(X + 32 768)2 (X − 8 000)2 , φ5 (X, X ) = −(X − 1 728)2 (X − 287 496)2 (X + 32 768)2 (X + 884 736)2 P1 , , φ6 (X, X ) = (X − 8 000)2 P12 P2 P32 P42 , φ7 (X, X ) = −(X + 3 375)(X − 16 581 375)X 2 (X − 54 000)2 × (X + 12 288 000)2 (X + 884 736)2 P22 , où P1 = X 2 − 1 264 000 X − 681 472 000, P2 = X 2 − 4 834 944 X + 14 670 139 392, P3 = X 2 + 191 025 X − 121 287 375, P4 = X 3 + 3 491 750 X 2 − 5 151 296 875 X + 12 771 880 859 375. Comme m n’est pas carré, les racines de φm (X, X ) dans C sont les invariants j (τ ) de courbes elliptiques a` multiplication complexe par un ordre quadratique imaginaire Z[τ ]. Les ordres quadratiques associés aux invariants racines des facteurs de degré 1 sont donnés dans la littérature (voir par exemple [Cohen 1993, 7.2.3]). En

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MARUSIA REBOLLEDO

déterminant les ordres quadratiques imaginaires possédant un e´ lément de norme 5, 6 et 7, on trouve une racine αi du facteur Pi de degré 2 pour i ∈ {1, . . . , 4}. On obtient ainsi les valeurs de Bi,i (m) (m ∈ {2, 3, 5, 6, 7}) données dans la table ci-dessous où a, b, c, d, e, f, g, h, v, w ∈ {0, 1} sont par définition e´ gaux a` 1 si et seulement si j (τ ) est supersingulier modulo p, c’est-à-dire si p est inerte ou ramifié dans Q(τ ). On supposera désormais que p > 173. Dans ce cas, les invariants apparaissant dans cette table sont tous distincts. L’égalité a = c e´ quivaut a` h = d et a = d e´ quivaut a` v = g. On fait parcourir au 8-uplet (a, b, c, d, e, f, g, w) les différentes valeurs possibles. Pour tous les 8-uplets distincts de ceux e´ numérés dans la Table 2, les fractions ι j (yk,m ), où k, m ∈ {2, 3, 5, 6, 7}, ne s’annulent pas simultanément. La table résume les résultats obtenus pour tous les 8-uplets posant un problème. Lorsque ι j (yk,m ) n’est pas une fraction identiquement nulle, on note n k,m le degré de son numérateur. Lorsque n k,m = 2, ι j (yk,m ) a un zéro dans Fp si et seulement si le discriminant dk,m de son numérateur (dans Z) est un carré modulo p.

ji 1728 287496 − 3375 16581375 8000 0 54000 − 12288000 − 32768 − 884736 α1 α2 α3 α4

τ √ −1 √ 2 −1 √ 1 2 (1 + −7) √ −7 √ −2 √ 1 2 (1 + −3) √ −3 √ 1 2 (1 + 3 −3) √ 1 (1 + −11) 2 √ 1 2 (1 + −19) √ −5 √ −6 √ 1 2 (1 + −15) √ 1 2 (1 + −23)

Bi (2) Bi (3) Bi (5) Bi (6) Bi (7)

Dτ 4

a

1

2

4

a

7

b

7

b

8

c

3

d

1

2

3

d

1

2

3

d

11

e

19

f

2

20

g

1

24

h

1

15

v

2

23 w

2

2 2

1 1

1

2

2

2 2

2 2 2

Table 1. Valeurs de Bi (m) = Bi,i (m), m ∈ {2, 3, 5, 6, 7}.

2


a b c d

e f

g w

Résultat

0 0 1 0

0 0

0 0

voir texte ci-dessous

0 0 0 0

0 0

0 ∗ 1 0

ι j (yk,m ) = 0 (k, m ∈ {2, 3, 5, 6, 7}) n 5,6 = 2, d5,6 = 26 .56 .115 .134 .372 .59.71, ι j (yk,m ) = 0 (k, m) 6= (5, 6) n 5,6 = 5, ι j (yk,m ) = 0 (k, m) 6= (5, 6)

1 1 0 1

0 0 0 1

ι j (yk,m ) = 0 (k, m ∈ {2, 3, 5, 6, 7}) ι j (y5,6 ) = ι j (y6,7 ) n 5,6 = 2, d5,6 = 26 .57 .74 .31.36319. p0 ι j (yk,m ) = 0 si (k, m) 6= (5, 6), (6, 7)

1 0

0 0 0 1

ι j (yk,m ) = 0 (k, m ∈ {2, 3, 5, 6, 7}) ι j (y3,6 ) = ι j (y5,6 ), n 5,6 = 2, d5,6 = 26 .3.57 .78 .112 .172 .192 .797 ι j (yk,m ) = 0 si (k, m) 6= (3, 6), (5, 6)

Table 2. Résultats des calculs pous les cas exceptionnels. Le symbole ∗ signifie que le coefficient peut prendre indifféremment la valeur 0 ou 1. On rappelle que p0 = 45321935159. ` titre d’exemple, lorsque (a, b, c, d, e, f, g, w) = (0, 0, 1, 0, 0, 0, 0, 0), on obA tient ι j (y2,3 ) = ι j (y2,5 ) = ι j (y3,5 ) = ι j (y5,6 ) = ι j (y5,7 ) = 0 et ι j (y2,6 ), ι j (y3,6 ), ι j (y2,7 ), ι j (y3,7 ) et ι j (y6,7 ) sont, a` multiplication par une puissance de 2 près, e´ gaux a` Q( j) =

P 0 ( j) −211 (13.181 j + 26 .33 .7.13.29) 2 − 2 = j − 8000 P2 ( j) ( j − 8000)P2 ( j)

qui s’annule en j0 ≡ −(181)−1 .26 .33 .7.29 mod p. Posons B l’ensemble des nombres premiers p qui sont simultanément un carré modulo 3, 4 et 7 et qui vérifient l’une des conditions suivantes : i) p carré modulo 5, 11, 19, et 23 et non carré modulo 8, ii) p est un carré modulo 5, 8, 11 et 19 ; iii) p carré modulo 8, 11 et 19, non carré modulo 5, 23 ; iv) p carré modulo 8, 11, 19, 23, 59, 71, non carré modulo 5 ; v) p carré modulo 8, 11, 19, 23, non carré modulo 5, 59, 71 ;

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MARUSIA REBOLLEDO

vi) p carré modulo 5, 8, 11, 23, non carré modulo 19 ;

vii) p carré modulo 5, 8, 11, non carré modulo 19, 23 et

p 31

p 36319

viii) p carré modulo 5, 8, 19, 23, non carré modulo 11 ;

p = 1; p0

ix) p carré modulo 5, 8, 19, 797, non carré modulo 11, 23. Lemme 4.1. Si p > 173, p 6= 797, 36319, p0 et p 6∈ B, alors pour tout j ∈ Fp non supersingulier, il existe (k, m) ∈ {2, 3, 5, 6, 7}2 tel que ι j (yk,m ) 6= 0 et par conséquent X 0+ ( pr )(Q) est trivial. Démonstration. On vérifie aisément que lorsque p est comme dans l’énoncé et j ∈ Fp , nous ne sommes pas dans l’un des cas e´ numérés dans la Table 2 et donc il y a une fraction non nulle parmi ι j (yk,m ), (k, m) ∈ {2, 3, 5, 6, 7}2 . Le lemme 4.1 et le théorème 1.1 de [Parent 2005] entraˆınent alors le Théorème 0.6. En effet, l’ensemble C du Théorème 0.6 est e´ gal a` A ∩ B et 797, 36319, p0 ne sont pas dans A (ici A est l’ensemble du théorème 1.1 de [Parent 2005]). Le cas des nombres premiers p = 11 et 17 ≤ p ≤ 173 a e´ té traité par Parent [2005, p. 8, preuve du théorème 1.1]. 1873 3217 7417 8233 9241 10333 11257 15733 16921 17389 18313 19273 21961 26161 26497 26833 30097 31081 32377 34057 35281 36793 38329 38833 41617 42337 42793 48409 Table 3. Nombres premiers p ≤ 50000 dans l’ensemble C.

Remerciements Je tiens ici a` remercier L. Merel, P. Parent ainsi que le referee pour leurs remarques durant l’élaboration de cet article. References [Arakawa et Böcherer 2003] T. Arakawa et S. Böcherer, “Vanishing of certain spaces of elliptic modular forms and some applications”, J. Reine Angew. Math. 559 (2003), 25–51. MR 2004m:11069 Zbl 1043.11040 [Bertolini et Darmon 1996] M. Bertolini et H. Darmon, “Heegner points on Mumford–Tate curves”, Invent. Math. 126:3 (1996), 413–456. MR 97k:11100 Zbl 0882.11034 [Bertolini et Darmon 1997] M. Bertolini et H. Darmon, “A rigid analytic Gross–Zagier formula and arithmetic applications”, Ann. of Math. (2) 146:1 (1997), 111–147. MR 99f:11079 Zbl 1029.11027 [Böcherer et Schulze-Pillot 1999] S. Böcherer et R. Schulze-Pillot, “Squares of automorphic forms on quaternion algebras and central values of L-functions of modular forms”, J. Number Theory 76:2 (1999), 194–205. MR 2000h:11046 Zbl 0940.11023


[Cohen 1993] H. Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics 138, Springer, Berlin, 1993. MR 94i:11105 Zbl 0786.11071 [Eichler 1955] M. Eichler, “Zur Zahlentheorie der Quaternionen-Algebren”, J. Reine Angew. Math. 195 (1955), 127–151. MR 18,297c Zbl 0068.03303 [Emerton 2002] M. Emerton, “Supersingular elliptic curves, theta series and weight two modular forms”, J. Amer. Math. Soc. 15:3 (2002), 671–714. MR 2003b:11038 Zbl 01739913 [Gross 1987] B. H. Gross, “Heights and the special values of L-series”, pp. 115–187 dans Number theory (Montreal, 1985), édité par H. Kisilevsky et J. Labute, CMS Conf. Proc. 7, Amer. Math. Soc., Providence, RI, 1987. MR 89c:11082 Zbl 0623.10019 [Gross et Kudla 1992] B. H. Gross et S. S. Kudla, “Heights and the central critical values of triple product L-functions”, Compositio Math. 81:2 (1992), 143–209. MR 93g:11047 Zbl 0807.11027 [Igusa 1959] J.-i. Igusa, “Kroneckerian model of fields of elliptic modular functions”, Amer. J. Math. 81 (1959), 561–577. MR 21 #7214 Zbl 0093.04502 [Illusie 1991] L. Illusie, “Réalisation l-adique de l’accouplement de monodromie d’après A. Grothendieck”, 196–197 (1991), 27–44. MR 93c:14020 Zbl 0781.14011 [Lang 1987] S. Lang, Elliptic functions, 2e éd., Graduate Texts in Mathematics 112, Springer, New York, 1987. MR 88c:11028 Zbl 0615.14018 [Merel 1996] L. Merel, “Bornes pour la torsion des courbes elliptiques sur les corps de nombres”, Invent. Math. 124:1-3 (1996), 437–449. MR 96i:11057 Zbl 0936.11037 [Mestre et Oesterlé ≥ 2008] J.-F. Mestre et J. Oesterlé, “Courbes elliptiques de conducteur premier”. Please supply year Non publié. [Miyake 1989] T. Miyake, Modular forms, Springer, Berlin, 1989. MR 90m:11062 Zbl 0701.11014 [Momose 1984] F. Momose, “Rational points on the modular curves X split ( p)”, Compositio Math. 52:1 (1984), 115–137. MR 86j:11064 Zbl 0574.14023 [Momose 1986] F. Momose, “Rational points on the modular curves X 0+ ( pr )”, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 33:3 (1986), 441–466. MR 88a:11056 Zbl 0621.14018 [Momose 1987] F. Momose, “Rational points on the modular curves X 0+ (N )”, J. Math. Soc. Japan 39:2 (1987), 269–286. MR 88h:14031 Zbl 0623.14009 [Ogg 1978] A. P. Ogg, “On the Weierstrass points of X 0 (N )”, Illinois J. Math. 22:1 (1978), 31–35. MR 57 #3136 Zbl 0374.14005 [Parent 2005] P. J. R. Parent, “Towards the triviality of X 0+ ( pr )(Q) for r > 1”, Compos. Math. 141:3 (2005), 561–572. MR 2006a:11076 Zbl 02183028 [Raynaud 1991] M. Raynaud, “Jacobienne des courbes modulaires et opérateurs de Hecke”, 196-197 (1991), 9–25. MR 93b:11077 Zbl 0781.14020 [Ribet 1983] K. A. Ribet, “Mod p Hecke operators and congruences between modular forms”, Invent. Math. 71:1 (1983), 193–205. MR 84j:10040 Zbl 0508.10018 [Serre 1972] J.-P. Serre, “Propriétés galoisiennes des points d’ordre fini des courbes elliptiques”, Invent. Math. 15:4 (1972), 259–331. Réimpression: pp. 1–73 dans ses Œuvres, tome 3, Springer, Berlin, 1986. MR 52 #8126 Zbl 0235.14012 [Serre 1986] J.-P. Serre, “Resume des cours 1975–1976”, pp. 284–291 dans ses Œuvres, vol. III, Springer, Berlin, 1986. MR 89h:01109c [de Shalit 1995] E. de Shalit, “On certain Galois representations related to the modular curve X 1 ( p)”, Compositio Math. 95:1 (1995), 69–100. MR 96i:11063 Zbl 0853.11045 [Zhang 2001] S.-W. Zhang, “Gross–Zagier formula for GL2 ”, Asian J. Math. 5:2 (2001), 183–290. MR 2003k:11101 Zbl 01818531

184

MARUSIA REBOLLEDO

Received December 1, 2006. Revised October 15, 2007. M ARUSIA R EBOLLEDO L ABORATOIRE DE M ATH E´ MATIQUES U NIVERSIT E´ B LAISE PASCAL C AMPUS UNIVERSITAIRE DES C E´ ZEAUX 63177 AUBI E` RE F RANCE [email protected] http://math.univ-bpclermont.fr/~rebolledo/


TWO REMARKS ON A THEOREM OF DIPENDRA PRASAD H IROSHI S AITO We show two results on local theta correspondence and restrictions of irreducible admissible representations of GL(2) over p-adic fields. Let F be a nonarchimedean local field of characteristic 0, and let L be a quadratic extension of F. Let L/F is the character of F × corresponding to the extension L/F, and let GL2 (F)+ be the subgroup of GL2 (F) consisting of elements with L/F (det g) = 1. The first result is that the theorem of Moen–Rogawski on the theta correspondence for the dual pair (U(1), U(1)) is equivalent to a result by D. Prasad on the restriction to GL2 (F)+ of the principal series representation of GL2 (F) associated with 1, L/F . As the second result, we show that we can deduce from this a theorem of D. Prasad on the restrictions to GL2 (F)+ of irreducible supercuspidal representations of GL2 (F) associated to characters of L × .

1. Introduction The purpose of this paper is to give two remarks on the comment in the last Remark in Section 3 of [Prasad 2007] and Theorem 1.2 in [Prasad 1994]. Let F be a nonarchimedean local field of characteristic 0, and let L be an quadratic extension of F. We denote by L/F the quadratic character of F × corresponding to the extension L/F. Let Ps(1, L/F ) be the normalized principal series representation of GL2 (F) associated to the characters 1 and L/F . We fix an embedding of L × into GL2 (F). The restriction of Ps(1, L/F ) to L × is a multiplicity-free direct sum. Let GL2 (F)+ be the subgroup of GL2 (F) consisting of elements with determinant belonging to N L/F (L × ). Then L × is contained in GL2 (F)+ , and the restriction of Ps(1, L/F ) to GL2 (F)+ decomposes into two irreducible subspaces Ps± (1, L/F ). In this situation, Lemma 4 in [Prasad 2007] states that a character φ of L × , whose restriction to F × is L/F , appears in Ps+ (1, L/F ) (resp. Ps− (1, L/F )) if and only if ε(φ, ψ0 ) = 1 (resp. −1). Here ψ0 is a character of L, the precise definition of which will be given in Section 3. On the other hand, we fix a character χ of L × whose restriction to MSC2000: primary 22E50, 11F27; secondary 11F70. Keywords: theta correspondence, epsilon factor. 185

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HIROSHI SAITO

F × is L/F , and consider the theta correspondence for the dual pair (U (1), U (1)) with respect to χ . Then the theorem of Moen–Rogawski states that a character η of L 1 appears in this theta correspondence if and only if ε(χη−1 L , ψ0 ) = 1 (see × [Moen 1987; Rogawski 1992]). Here η L is the character of L given by η L (x) = η(x/x) ¯ for x ∈ L × . Now the correspondence η 7→ χη−1 L yields a one to one correspondence 1 between characters of L and characters of L × whose restriction to F × is L/F . Thus the factor ε(φ, ψ0 ) appears in formulas expressing characters of linear and nonlinear groups. The Remark in Section 3 of [Prasad 2007] raises the question whether there is a natural explanation for this phenomenon. Our first remark is an answer to this question. Our result is that Lemma 4 in Prasad’s article is equivalent to the theorem of Moen–Rogawski. We show this in Sections 3 and 4 using seesaw diagrams after some preparations on seesaw diagrams in Section 2. We note that both the theorem of Moen–Rogawski and Prasad’s Lemma 4 were originally proved by local methods for F with odd residual characteristic, and the general cases were proved by these local results and global methods (see [Moen 1987], Proposition 3.4 of [Rogawski 1992], and Lemma 4 of [Prasad 2007]). Later a purely local proof for the theorem of Moen–Rogawski was given by Harris, Kudla and Sweet (see Corollaries 8.5 and A.9 of [Harris et al. 1996]), and that of Lemma 4 of [Prasad 2007] was given by the author (see Appendix of [Prasad 2007]). The second remark is concerned with Theorem 1.2 in [Prasad 1994]. Let π be the irreducible supercuspidal representation of GL2 (F) associated to a character λ of L × by theta correspondence. Then π| L × is multiplicity-free, and π|GL2 (F)+ decomposes into two irreducible subspaces π + and π − . In the article in question, D. Prasad proved that φ with λφ −1 | F × = L/F appears in π ± if and only if ¯ −1 , ψ0 ) = ±1. In Section 3 we deduce an analogue of this ε(λφ −1 , ψ0 ) = ε(λφ theorem for unitary groups of degree 2 (Theorem 3.5) from the theorem of Moen– Rogawski using a seesaw diagram. In Section 4 we show the above theorem of D. Prasad from this again using a seesaw diagram, which is found in [Harris 1993]. This is the first half of Theorem 1.2 in [Prasad 1994]. In Section 5, we treat a similar problem for representations of multiplicative group of the division quaternion algebra. This is the second half of Theorem 1.2 in [Prasad 1994]. 2. Seesaw diagrams In this section, we introduce notation and recall some seesaw diagrams which will be used in later sections. Let F, L and L/F be as before, and fix a nontrivial additive character ψ of F. For α ∈ L, we denote by α¯ its conjugate over F. We fix δ ∈ L × such that δ¯ = −δ and n 0 ∈ F × not contained in N L/F (L × ).

TWO REMARKS ON A THEOREM OF DIPENDRA PRASAD

187

For a finite-dimensional L-space W equipped with hermitian or antihermitian form, we denote by U (W ) its unitary group and by GU(W ) its unitary similitude group. For a vector space W with symplectic form, we denote by Sp(W) its symplectic group and by GSp(W) its symplectic similitude group. We denote by Mp(W) the metaplectic group of W. Let V 0 be a finite-dimensional right Fspace with symmetric bilinear form hv, v 0 i F for v, v 0 ∈ V 0 . We denote by SO(V 0 ), O(V 0 ), and GO(V 0 ) the special orthogonal group, the orthogonal group, and the orthogonal similitude group of V 0 respectively. We denote by GO+ (V 0 ) the group of proper similitudes of V 0 . Let V be a finite-dimensional right L-space with hermitian form satisfying hv1 α, v2 βi = αhv ¯ 1 , v2 iβ,

v1 , v2 ∈ V

and let W be a left L-space with antihermitian form satisfying ¯ hαw1 , βw2 i = αhw1 , w2 iβ,

w1 , w2 ∈ W

for α, β ∈ L. Then on W = V ⊗ L W , we can define a symplectic form by hhv1 ⊗ w1 , v2 ⊗ w2 ii = 21 tr L/F hv1 , v2 ihw1 , w2 i . For W, V , we have a dual reductive pair (U (W ), U (V )) in Sp(W). We denote the natural embeddings by ιV : U (W ) → Sp(W), ιW : U (V ) → Sp(W). Assume W is a direct sum of two antihermitian spaces W1 , W2 for L/F, and set

Wi = V ⊗ Wi for i = 1, 2. Similarly as above, we have dual pairs (U (W1 ), U (V )) in Sp(W1 ) and (U (W2 ), U (V )) in Sp(W2 ), and the embeddings

ιV,1 : U (W1 ) → Sp(W1 ), ιW1 : U (V ) → Sp(W1 ), ιV,2 : U (W2 ) → Sp(W2 ), ιW2 : U (V ) → Sp(W2 ). These dual pairs yield the seesaw diagram

(2-1)

U (W )Q

U (V ) × Sp(W2 )

U (W1 ) × {1}

U (V )

QQQ m QQQ mmmmm Q m mmQQ mmm QQQQQ mmm Q

The right vertical line is the map ιW1 × ιW2 : U (V ) → U (V ) × Sp(W2 ) ⊂ Sp(W1 ) × Sp(W2 ).

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We recall one more seesaw diagram from [Harris 1993]. Let W 0 be a finitedimensional left F-space with symplectic form h , i F . We can define an antihermitian form on W L0 = L ⊗ F W 0 by X X X 0 αi ⊗ vi , βj ⊗ vj = αi β¯ j hvi , v 0j i F i

j

i, j

for αi , β j ∈ L, and vi . v j ∈ V 0 . Conversely, let V be a right L-space with hermitian form h , i. Then composing the hermitian form with tr L/F , we can define a symmetric bilinear form 0 1 2 tr L/F (hv, v i) on Res F V , the space V considered as an F-space. In this notation we have, from [Harris 1993, (3.5.1.1)],

(2-2)

GU(W L0 )

GO(Res F V )

GSp(W 0 )

GU(V )

QQQ mmm QQQ QQmQmmmm m QQQ mmm QQQ mmm

3. Application of the theorem of Moen–Rogawski In this section, using the diagram (2-1) and the theorem of Moen–Rogawski, we deduce an analogue of Theorem 1.2 in [Prasad 1994] for unitary groups of degree 2. For α ∈ L × with α¯ = −α, we denote by W (α) the 1 dimensional left L-space L with antihermitian form hx, yi = αx y¯ for x, y ∈ L. For α, β ∈ L × , we set W (α, β) = W (α)⊕ W (β). For a ∈ F × , we denote by V (a) the 1 dimensional right L-space L with hermitian form hx, yi = a x¯ y. We set W = W (δ), W− = W (−δ), and V = V (1), or W = W (n 0 δ), W− = W (−n 0 δ), and V = V (1). Set W = V ⊗ L W , and W− = V ⊗ L W− . Then we have a seesaw diagram of type (2-1): U (W + W− )

U (V ) × Sp(W− )

U (W ) × {1}

U (V )

QQQ QQQ mmm QQmQmmmm m Q mmm QQQQQ mmm Q

We recall the splittings of the above unitary groups into metaplectic groups, following Section 1 of [Harris et al. 1996]. We fix a character χ of L × whose restriction to F × is L/F . Let X be the graph of minus the identity from W to W− , and let Y be the graph of the identity. Then V ⊗ L X and V ⊗ L Y are maximal isotropic subspace of W, and W = V ⊗ L X +V ⊗ L Y yields a complete polarization


189

of W. This determines an isomorphism Mp(W + W− ) ' Sp(W + W− ) o C1 , where the product in Sp(W + W− ) o C1 is given by the Rao cocycle [1993]. The inverse image in Mp(W + W− ) of Sp(W) × {1} or {1} × Sp(W) is isomorphic to Mp(W). By (1.21) of [Harris et al. 1996], we have splittings ι˜V,χ , ι˜V,χ × ι˜V,χ ,− satisfying ι˜V,χ

U (W + W− ) i

- Mp(W + W− ) 6

6

i˜

U (W ) × U (W )

ι˜V,χ ×˜ιV,χ ,−

- Mp(W) × Mp(W).

Here we note that U (W− ) = U (W ), Mp(W) = Mp(W− ) and the splitting i˜ : Mp(W) × Mp(W) → Mp(W + W− ) of the embedding i : Sp(W) × Sp(W) → Sp(W + W− ) is specified so that the restriction to central C1 is given by C1 × C1 → C1 ,

(c1 , c2 ) → c1 c¯2 .

Then, by [Harris et al. 1996, Lemma 1.1], (3-1)

ι˜V,χ ,− = χ −1 ι˜V,χ .

In this case, U (V ) is the center of U (W + W− ), and the splitting of U (V ) as the center of U (W + W− ) by χ coincides with the splitting ιW +W− ,χ 2 (Corollary A.8 of the same reference). Let (ωψ , S(V ⊗ L X )) be the Weil representation of Mp(W + W− ) realized on the space of Schwartz–Bruhat functions on V ⊗ F X as the Schrödinger model associated to the complete polarization W = V ⊗ L X + V ⊗ L Y . For a character λ1 of U (V ), let θχ (λ1 , W + W− ) be the theta correspondence of λ1 to U (W + W− ). Namely, let SV,W,χ (λ1 ) be the maximal quotient of S(V ⊗ L X ) on which U (V ) acts as multiple of λ1 . Then SV,W,χ (λ1 ) ' θχ (λ1 , W + W− ) λ1 , as U (W + W− ) × U (V )-spaces with an U (W + W− )-module θχ (λ1 , W + W− ). Let ωψ.W be the Weil representation of Mp(W). Let ψ0 be the additive character of L given by ψ0 (x) = ψ( 21 tr L/F (−δx)) for x ∈ L. For a character η of L 1 , we denote by η L the character of L × given by η L (x) = η(x/x). ¯

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Theorem 3.1 (Moen and Rogawski). Let 1 if W = W (δ), = −1 if W = W (n 0 δ). Then ωψ,W ◦ ι˜V,χ |U (W ) =

M

Cη.

ε(χ η−1 L ,ψ0 )=

Remark 3.2. Here we use the character ψ0 instead of ψ ◦ tr L/F . This simplifies some expressions (see Remark in Introduction of [Prasad 1994]). For a character η of U (W ), we denote by θχ (η, V ) the theta correspondence of η in Mp(W) to U (V ). Then θχ (η, V ) = η−1 if η appears in the theta correspondence. We note that U (V ) ' U (W ) ' L 1 , and the embedding ιV and ιW are chosen so that the actions of U (V ) and U (W ) on W are the inverse of each other. By the isomorphism U (V ) ' L 1 , we consider the restriction of χ to L 1 as a character of U (V ) and denote it also by χ. Lemma 3.3. Let the notation be as above. Let U (W ) × {1} be the subgroup of U (W ) × U (W )(⊂ U (W + W− )) consisting of elements with unit in the second component. Then dim HomU (W )×{1} θχ (χ −1 λ1 , W + W− ), η 1 1 if η and λ1 η appear in ωψ,W ◦ ι˜V,χ , = 0 otherwise. Proof. Hom(U (W )×{1})×U (V ) (ωψ , (η 1) χ −1 λ1 ) ' Hom(U (W )×{1})×U (V ) (θχ (χ −1 λ1 , W + W− ) χ −1 λ1 , (η 1) χ −1 λ1 ) ' HomU (W )×{1} (θχ (χ −1 λ1 , W + W− ), η 1). We note that U (V ) is embedded into U (W ) × U (W ) diagonally in Sp(W + W− ), and the action of α ∈ U (V ) for α ∈ L 1 on S(V ⊗ L X ) is given by that of (α −1 , α −1 ) ∈ U (W )×U (W ). By [Mœglin et al. 1987, II.1, Remarques (5), (6)] and [Harris et al. ∨ 1996, Lemma 2.1(i)], the restriction of ωψ to Mp(W) × Mp(W) is ωψ,W ωψ, W. ∨ Here ωψ,W is the contragredient of ωψ,W , and by (3-1) we obtain ∨ ∨ ωψ, W ◦ ι˜V,χ ,− = χ(ωψ,W ◦ ι˜V,χ ) .

Hence Hom(U (W )×{1})×U (V ) (ωψ , (η 1) χ −1 λ1 ) ∨ −1 1 ' HomU (W )×U (V ) η (θχ (η, V ) ⊗ ωψ, λ W ◦ ι˜V,χ ,− ), η χ ∨ −1 1 ' HomU (V ) (θχ (η, V ) ⊗ (χ(ωψ,W ◦ ι˜V,χ , ) ), χ λ .


191

Our assertion follows from this.

Taking λ1 to be the trivial character of L 1 , by Lemma 3.3 and Theorem 3.1, we obtain: Theorem 3.4. Let be as above. Then θχ (χ −1 , W + W− )|U (W )×{1} =

M

Cη 1.

ε(χ η−1 L ,ψ0 )=

Theorem 3.5. Let λ1 be a nontrivial character of L 1 , and let be as above. Then M θχ (χ −1 λ1 , W + W− )|U (W )×{1} = Cη 1. ε(χ (λ1L η L )−1 ,ψ0 ) = ε(χ η−1 L ,ψ0 ) =

4. Prasad’s Theorem We rewrite the results in the previous section in terms of GU(2) and a torus TL in GU(2) isomorphic to L × , and by restricting it to a subgroup of index 2 of GL2 (F), we deduce the theorem of D. Prasad using a seesaw diagram of type (2-2). Let W 0 = F 2 be the two-dimensional left F-space with symplectic form hv1 , v2 i = x1 y2 − y1 x2 for v1 = (x1 , y1 ), v2 = (x2 , y2 ) ∈ W 0 , and let W L0 = L 2 be the two-dimensional left L-space with antihermitian from hv˜1 , v˜2 i H = x1 y¯2 − y1 x¯2 for v˜1 = (x1 , y1 ), v˜2 = (x2 , y2 ) ∈ W L0 . Then we see W (δ, −δ) ' W L0 , as spaces with antihermitian forms. More explicitly, let δ 1/2 h= . −δ 1/2 Then

0 1 h −1 0

t

¯h = δ 0 . 0 −δ

If we take n 0 hv1 , v2 i instead of hv1 , v2 i, we get W (n 0 δ, −n 0 δ) ' W L0 . Similarly, we have 0 n0 t ¯ n δ 0 h h= 0 . −n 0 0 0 −n 0 δ Let Res F V be the two-dimensional right F-space with symmetric bilinear form associated with V (1). For these spaces, we have the following diagram of type

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HIROSHI SAITO

(2-2): GU(W L0 )

GO(Res F V )

GSp(W 0 )

GU(V )


Note that

W0

and V satisfy

GSp(W 0 ) = GL(W 0 ),

Sp(W 0 ) = SL(W 0 ),

SO(Res F V ) = U (V ),

U (W L0 ) ⊃ SU (W L0 ) = SL(W 0 ),

GO+ (Res F V ) = GU(V ).

Let νW (g) be the similitude of g ∈ U (W L0 ). Let GU(W L0 )+ = {g ∈ GU(W L0 ) | L/F (νW (g)) = 1}, GL(W 0 )+ = {g ∈ GL(W 0 ) | L/F (det g) = 1}, and identify L × with the center of GU(W L ). Then GU(W L0 ) ⊃ GU(W L0 )+ = L ×U (W L0 ) = L × GL(W 0 )+ , since N L/F (L × )L ×2 = L 1 L ×2 . Let TL be the torus in GL(W 0 ) isomorphic to L × given by a 2−1 b t 2 t (a, b) ∈ F \ (0, 0) , 2δ 2 b a and let α = a + bδ ∈ L ,

µ = α/α. ¯

We fix the isomorphism a 2−1 b α 7→ 2δ 2 b a

and identify TL with L × . We have µ+1 (2δ)−1 (µ − 1) a 2−1 b α¯ 0 1 . (4-1) = µ+1 2δ 2 b a 0 α¯ 2 (2δ)(µ − 1) We note

1 µ + 1 (2δ)−1 (µ − 1) µ 0 = h −1 h. µ+1 0 1 2 2δ(µ − 1)

We recall the action of some elements on S(V ⊗ L X ). We write them for the pair (U (W L0 ), U (V )). Then X = {(x, 0) | x ∈ L}, Y = {(0, y) | y ∈ L}, and α 0 ∈ U (W L0 ) 0 α¯ −1


193

acts on X by α and on Y by α¯ −1 . Hence α 0 = χ(α¯ −1 ) = χ(α) βV 0 α¯ −1 in the notation of Theorem 3.1 of [Kudla 1994]. By the same theorem we have, for α ∈ L × , α 0 1/2 f (x) = χ(α)|α| L f (αx). ωψ ι˜V,χ 0 α¯ −1 In particular, for α ∈ L 1 , α 0 (4-2) ωψ ι˜V,χ f (x) = χ(α) f (αx). 0 α For the dual pair (SL(W 0 ), SO(Res F V )), let S(λ1 ) be the maximal quotient of S(V ⊗ L X ) = S(Res F V ⊗ F X 0 ), X 0 = {(x, 0) | x ∈ F}, on which SO(Res F V ) acts as multiple of λ1 . Here the action of α ∈ SO(Res F V ) with α ∈ L 1 is given by f (x) 7→ f (α −1 x). Then the above formula implies that SV,W,χ (χ −1 λ1 ) = S(λ1 ). Hence the restriction of the action of U (W L0 ) on the space θχ (χ −1 λ1 , W + W− ) to SL(W 0 ) is the theta correspondence of λ1 to SL(W 0 ). We denote it by θ(λ1 , W 0 ). We extend the theta correspondence θχ of U (V ) to U (W L0 ) to that of GU(V ) to GU(W L0 )+ following [Harris 1993, 3.2]. The similitude νV of GU(V ) satisfies νV (GU(V )) = N L/F L × . Let R(V, W ) = {(g, h) ∈ GU(W L0 ) × GU(V ) | νV (h) = νW (g)}. Then by corresponding (g, h) to the map v ⊗ w 7→ h −1 v ⊗ wg, v ∈ V, w ∈ W L0 , we can takes R(V, W ) into Sp((V ⊗ L W L0 ). We consider a semidirect product U (W L0 ) n GU(V ) defined by hg = hgh with −1 1 0 1 0 . g= g 0 νV (h) 0 νV (h)

h

Then we have an isomorphism R(V, W ) ' U (W L0 ) n GU(V )

194

HIROSHI SAITO

given by −1 1 0 ,h . (g, h) → g 0 νV (h) We let GU(V ) act on S(V ⊗ L X ) by −1/2

L(h) f (x) = χ(α −1 )|α| L

f (α −1 x).

Then L(h) defines a unitary operator on S(V ⊗ L X ), and this action with ωψ ◦ ι˜V,χ defines an action of R(V, W ) on S(V ⊗ L X ) and a splitting of R(V, W ) into Mp(V ⊗ L W L0 ). Let λ be a character of GU(V ) whose restriction to U (V ) is λ1 . We identify λ with a character of L × by GU(V ) ' L × . For a character λ of L × , let λ be the ¯ for α ∈ L × . By the projection to the second character of L × given by λ(α) = λ(α) 0 factor GU(V ) of GU(W L ) × GU(V ), we may see χλ as a character of R(V, W ). Define (S(V ⊗ L X ) ⊗ χ λ)U (V ) to be the maximal quotient of S(V ⊗ L X )⊗χ λ on which U (V ) acts trivially. Then GU(W L0 )+ acts on this space as follows. For g ∈ GU(W L0 )+ , choose h ∈ GU(V ) satisfying νW (g) = νV (h). Define the action of g as that of (g, h) ∈ R(W, V ). Then this is independent of the choice of h. As U (W L0 )-modules, we have (S(V ⊗ L X ) ⊗ χλ)U (V ) ' SV,W,χ (χ −1 λ1 ), and on this space, α0 α0 ∈ GU(W L0 )+ acts by χ λ. We denote the restriction to GL(W 0 )+ of this representation by θ(λ, GL(W 0 )) . Here is as in Section 3. Let a = α α. ¯ Then a 0 α¯ 0 α 0 = 0 1 0 α¯ 0 α¯ −1 Hence α0 01 acts on f˜ ∈ S(λ1 ) sending it to the class in S(λ1 ) of the function 1/2

1/2

χ (α)λ(α)χ ¯ (α)|α| L f (αx) = λ(α)|α| L f (αx). This coincides with the extension of the action of SL(W 0 ) to GL(W 0 )+ in [Jacquet and Langlands 1970, Proposition 1.5]. For a character λ of L × , we set ) 0 + + θ (λ, GL(W 0 )) = Ind GL(W GL(W 0 )+ θ(λ, GL(W ) ) . 0

Then as GL(W 0 )+ -modules, we have ) 0 0 + + 0 + − ResGL(W GL(W 0 )+ θ (λ, GL(W )) = θ(λ, GL(W ) ) ⊕ θ(λ, GL(W ) ) ; 0

see [Mœglin et al. 1987, II.1, Remarque (3)]. Let Ps(1, L/F ) be the principal series representation of GL2 (F) associated with characters (1, L/F ). Then θ (1, GL(W 0 )) is isomorphic to Ps(1, L/F ) by [Jacquet


195

and Langlands 1970, Theorem 4.7]. We set Ps(1, L/F ) the subspace corresponding to θ (λ, GL(W 0 )) . By setting φ = χη−1 L , we see that Theorem 3.4 is equivalent to the following: Theorem 4.1 [Prasad 2007, Lemma 4]. For = ±1, Ps(1, ω L/F ) |TL = ⊕ε(φ,ψ0 )= Cφ, where φ runs through all characters of L × whose restriction to F × is equal to L/F . Remark 4.2. The map η 7→ χ −1 η L induces a one to one correspondence between the set of characters of L 1 and the set of characters of L × whose restriction to F × is L/F . Therefore the theorem of Moen–Rogawski is equivalent to the preceding theorem through Theorem 3.4. For λ such that λ| L 1 is not trivial, θ(λ, GL(W 0 )) is an irreducible supercuspidal representation of GL(W 0 ) by Theorem 4.6 of [Jacquet and Langlands 1970]. In this case, Theorem 3.5 can be stated as follows: Theorem 4.3 [Prasad 1994, Theorem 1.2]. Under the action of GL2 (F)+ , the space θ (λ, GL(W 0 )) decomposes into two subspaces θ(λ, GL(W 0 ))± , and for = ±1, one has M θ (λ, GL(W 0 )) |TL = Cφ, ε(λφ −1 ,ψ0 ) = ε(λφ −1 ,ψ0 ) =

where φ runs through all characters of L × which satisfy λφ −1 | F × = L/F . Proof. Set φ = χ −1 λη L . Since λ1L = λλ−1 , we see −1 χ (η L λ1L )−1 = (χλ−1 η−1 L )λ = λφ , −1 −1 −1 χη−1 L = (χλ η L )λ = λφ .

We note λφ −1 | F × = λφ −1 | F × = L/F . By (4-1), we can see the action of TL by that of L 1 . For v ∈ θχ (χ −1 λ1 , W + W− ), U (W ) × {1} acts on v via η 1 if and only if TL acts on v via χ¯ λη L = χ −1 λη L . The assertion follows from this and Theorem 3.5. 5. Nonsplit case We now consider the nonsplit case. Let α β B= α, β ∈ L . n 0 β¯ α¯

196

HIROSHI SAITO

Then B is the division quaternion algebra over F. Let B + = {x ∈ B | L/F (N (x)) = 1},

B 1 = {x ∈ B | N (x) = 1}.

Here N (x) is the reduced norm of x ∈ B. We set α 0 × TL = α ∈ L . 0 α¯ Then TL ' L × . We note (5-1)

α 0 α¯ 0 α/α¯ 0 = . 0 α¯ 0 α¯ 0 1

Let α = δ, β = −n 0 δ, or α = n 0 δ, β = −n 20 δ. Then B × ⊂ GU(W (α, β)), and TL ⊂ B + ⊂ GU(W (α, β))+ = L ×U (W (α, β)) = L × B + . Here GU(W (α, β))+ is the subgroup of GU(W (α, β)) consisting of elements with similitude in N L/F (L × ). We define splittings. Let W = W (α, −β). We embed W (α, β) into W + W− and consider U (W (α, β)) as a subgroup of U (W + W− ). Let W = V ⊗ F W , and W− = V ⊗ F W− . We may consider W(α, β) = V ⊗ F W (α, β) as a symplectic subspace of W + W− and Sp(W(α, β))) as a subgroup of Sp(W + W− ). Then we have splittings ι˜V,χ , ι˜V,χ,− satisfying ι˜V,χ

U (W + W− ) i

- Mp(W + W− ) 6

6

U (W ) × U (W )

i˜

ι˜V,χ × ι˜V,χ ,−

- Mp(W) × Mp(W).

We choose the embedding of Mp(W)×Mp(W) into Mp(W +W− ) so that it induces the map (c1 , c2 ) 7→ c1 c¯2 on the center C1 × C1 . Let W(α) = V ⊗ L W (α), and W(β) = V ⊗ L W (β). Restricting the above diagram to Mp(W(α, β)), we obtain U (W (α, β)) i

ι˜V,χ

- Mp(W(α, β)) 6

6

U (W (α)) × U (W (β))

i˜


- Mp(W(α)) × Mp(W(β))

Here Mp(W(α, β)) is the inverse image of Sp(W(α, β)) in Mp(W + W− ), and Mp(W(α)) and Mp(W(β)) are the inverse images of Sp(W(α)) and Sp(W(β)) in Mp(W) on the first and the second factor in the above diagram respectively. The


197

restriction of ι˜V,χ : U (W (α, −β)) → Mp(W) to U (W (−β)) induces a map U (W (−β)) = U (W (β))

ι˜V,χ

- Mp(W(−β)) = Mp(W(β)).

Then ι˜V,χ,− and χ −1 ι˜V,χ coincide as homomorphisms of U (W (β)) to Mp(W(β)), by [Harris et al. 1996, Lemma 1.1]. We have ∨ ωψ,W+W ◦ i˜ = ωψ,W(α,−β) ωψ, W(α,−β)

by [Mœglin et al. 1987, II.1 Remarques (5), (6)] and [Harris et al. 1996, Lemma 2.1(i)]. By restricting this to Mp(W(α)) × Mp(W (β)), we obtain ∨ ωψ,W(α,β) ◦ i˜ = ωψ,W(α) χωψ, W(−β) , ∨ ωψ,W(α,β) ◦ i˜ ◦ (˜ιV,χ × ι˜V,χ ,− ) = ωψ,W(α) ◦ ι˜V,χ ωψ, W(−β) ◦ ι˜V,χ ,− ∨ = ωψ,W (α) ◦ ι˜V,χ χωψ,W (−β) ◦ ι˜V,χ

As for the splitting for U (V ), we may take ι˜W +W− ,χ 4 or that induced by ι˜V,χ . Let θχ (χ −1 λ1 , W (α, β)) be the theta correspondence of the character χ −1 λ1 of U (V ) to U (W (α, β)) in Mp(W(α, β)). By the same calculation as in the split case, we obtain: Lemma 5.1. Let U (W (α)) × {1} be the subgroup of U (W (α)) × U (W (β)). Then dim HomU (W (α))×{1} θχ (χ −1 λ1 , W (α, β)), η 1 1 if η appears in ωψ,W(α) ◦ ι˜V,χ and λ1 η appears in ωψ,W(−β) ◦ ι˜V,χ , = 0 otherwise. Since L/F (−β/α) = −1, the trivial character does not satisfy the above condition for λ1 . In the case of a nontrivial λ1 , we have: Theorem 5.2. Let λ1 be a nontrivial character of L 1 , and let = L/F (α/δ). Then M θχ (χ −1 λ1 , W (α, β))|U (W (α))×{1} = Cη 1. −ε(χ (λ1L η L )−1 λ1L ,ψ0 ) = ε(χ η−1 L ,ψ0 ) =

As in the split case, we can interpret this result by the dual reductive pair (B × , GO(V )). In the same way as in the split case, we can define θ(λ1 , B 1 ). Let λ be a character of L × which restriction to L 1 is λ1 . We define the action of L × , the center of U (W (α, β)), on θχ (χ −1 λ1 , W (α, β)) by χλ. Then this yields a well-defined smooth action of L ×U (W (α, β)) on θχ (χ −1 λ1 , W (α, β)), since L × ∩ U (W (α, β)) = L 1 . By restriction, we obtain an action of B + , since B + ⊂ L ×U (W (α, β)). We denote this representation of B + by θ(λ, B + ) for = L/F (α/δ). We induce it to B × and denote it by θ(λ, B × ). By Theorem 5.2 and (5-1), we obtain:

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HIROSHI SAITO

Theorem 5.3. Under the action of B + , θ(λ, B × ) decomposes into two subspaces θ (λ, B × ) for = ±1, and M θ (B × , λ) |TL = Cφ, −ε(λφ −1 ,ψ0 )= ε(λφ −1 ,ψ0 )=

where φ runs through all characters of L × that satisfy λφ −1 | F × = L/F . Remark 5.4. The representations θ(λ, GL(W 0 )) and θ(λ, B × ) are in Jacquet– Langlands correspondence with each other, and Theorem 5.3 gives the latter half of Theorem 1.2 in [Prasad 1994]. By [Mœglin et al. 1987, Chapitre 3, IV, Corollaire 9], an irreducible quotient of θ χ −1 λ1 , W (U (α, β)) is uniquely determined. Since U (W (α, β)) is compact, θ(χ −1 λ1 , U (W (α, β))) is a multiple of this irreducible representation. Lemma 5.1 implies that the multiplicity is 1, and θ (χ −1 λ1 , W (α, β)) is irreducible. Let π = θ(λ, B × ). Since λ| L 1 is not trivial, θ (λ, GL(W 0 )) is supercuspidal. Let π 0 be the representation of B × which corresponds to θ (λ, GL(W 0 )) under the Jacquet–Langlands correspondence. We denote by χπ , χπ 0 the characters of π, π 0 . Then π and π 0 satisfy π ⊗ L/F ' π,

π 0 ⊗ L/F ' π 0 ,

and χπ = χπ 0 on L × . By Corollaries 1.7 and 1.15 of [Hijikata et al. 1993] and Theorem 4.6 (and the remark following it) in [Takahashi 1996], this implies that χπ = χπ 0 on all the other elliptic torus of B × . Therefore π ' π 0 . Acknowledgement The author thanks Professor D. Prasad for calling his attention to these problems. References [Harris 1993] M. Harris, “L-functions of 2 × 2 unitary groups and factorization of periods of Hilbert modular forms”, J. Amer. Math. Soc. 6:3 (1993), 637–719. MR 93m:11043 Zbl 0779.11023 [Harris et al. 1996] M. Harris, S. S. Kudla, and W. J. Sweet, “Theta dichotomy for unitary groups”, J. Amer. Math. Soc. 9:4 (1996), 941–1004. MR 96m:11041 Zbl 0870.11026 [Hijikata et al. 1993] H. Hijikata, H. Saito, and M. Yamauchi, “Representations of quaternion algebras over local fields and trace formulas of Hecke operators”, J. Number Theory 43:2 (1993), 123–167. MR 94e:11126 Zbl 0819.11018 [Jacquet and Langlands 1970] H. Jacquet and R. P. Langlands, Automorphic forms on GL(2), Lecture Notes in Mathematics 114, Springer, Berlin, 1970. MR 53 #5481 Zbl 0236.12010 [Kudla 1994] S. S. Kudla, “Splitting metaplectic covers of dual reductive pairs”, Israel J. Math. 87:1-3 (1994), 361–401. MR 95h:22019 Zbl 0840.22029


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[Mœglin et al. 1987] C. Mœglin, M.-F. Vignéras, and J.-L. Waldspurger, Correspondances de Howe sur un corps p-adique, Lecture Notes in Mathematics 1291, Springer, Berlin, 1987. MR 91f:11040 Zbl 0642.22002 [Moen 1987] C. Moen, “The dual pair (U(3), U(1)) over a p-adic field”, Pacific J. Math. 127:1 (1987), 141–154. MR 88e:22025 Zbl 0675.22008 [Prasad 1994] D. Prasad, “On an extension of a theorem of Tunnell”, Compositio Math. 94:1 (1994), 19–28. MR 95k:22023 Zbl 0824.11035 [Prasad 2007] D. Prasad, “Relating invariant linear form and local epsilon factors via global methods”, Duke Math. J. 138:2 (2007), 233–261. MR 2318284 Zbl 05170260 [Ranga Rao 1993] R. Ranga Rao, “On some explicit formulas in the theory of Weil representation”, Pacific J. Math. 157:2 (1993), 335–371. MR 94a:22037 Zbl 0794.58017 [Rogawski 1992] J. D. Rogawski, “The multiplicity formula for A-packets”, pp. 395–419 in The zeta functions of Picard modular surfaces (Montreal, 1988), edited by R. P. Langlands and D. Ramakrishnan, Publications CRM, Montreal, 1992. MR 93f:11042 Zbl 0823.11027 [Takahashi 1996] T. Takahashi, “Character formula for representations of local quaternion algebras (wildly ramified case)”, J. Math. Kyoto Univ. 36:1 (1996), 151–197. MR 97f:11096 Zbl 0897.22018 Received March 2, 2007. H IROSHI S AITO D EPARTMENT OF M ATHEMATICS FACULTY OF S CIENCE K YOTO U NIVERSITY K YOTO 606-8502 JAPAN [email protected]


MODULE SUPERSINGULIER, FORMULE DE GROSS–KUDLA ET POINTS RATIONNELS DE COURBES MODULAIRES M ARUSIA R EBOLLEDO We show how the Gross–Kudla formula about triple product L-functions allows us to construct degree-zero elements of the supersingular module annihilated by the winding ideal. Using the method of Parent, we apply those results to the study of rational points on modular curves, determining a set of primes of analytic density 1−9/210 for which the quotient of X 0 ( p r ) (r > 1) by the Atkin–Lehner operator w p r has no rational points other than the cusps and the CM points.

Introduction Pour N > 0 un entier, notons X 0 (N ) la courbe modulaire sur Q classifiant grossièrement les courbes elliptiques généralisées munies d’une N -isogénie. La motivation initiale de ces travaux est l’étude des points rationnels du quotient X 0+ ( pr ) de X 0 ( pr ) par l’involution d’Atkin–Lehner w pr pour p un nombre premier et r > 1 un entier. Pour cela, nous reprenons une méthode de Parent [2005] s’inspirant des travaux de Momose [1984; 1986; 1987] et faisant appel au module supersingulier. Fixons p > 3 un nombre premier et F¯ p une clôture algébrique de Fp . Nous appelons module supersingulier le Z-module libre P engendré par l’ensemble fini S = {x 0 , . . . , x g } des classes d’isomorphismes de courbes elliptiques supersingulières sur F¯ p . Notons P0 le sous-groupe de P constitué des e´ léments de degré nul. On peut munir P0 d’une action de l’anneau T engendré par les opérateurs de Hecke agissant sur l’espace vectoriel S2 (00 ( p)) des formes paraboliques de poids 2 pour 00 ( p) (voir 1A). Notons P0 [Ie ] l’ensemble des e´ léments de P0 annulés par l’idéal d’enroulement Ie ⊂ T c’est-à-dire l’annulateur des formes primitives f ∈ S2 (00 ( p)) telles que L( f, 1) 6= 0. Pour j ∈ P1 (F¯ p ) un invariant non supersingulier, considérons l’application (1)

ιj :

P → F¯ p

Pg

i=0 λi x i

7→

Pg

i=0

λi j − ji

MSC2000: primary 14G05, 11G05, 11G18; secondary 14G10, 11R52. Mots-clefs: rational points on modular curves, supersingular module, special values of L-functions. 223

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MARUSIA REBOLLEDO

où, pour i ∈ {0, . . . , g}, ji est l’invariant d’une courbe elliptique E i ∈ xi . Parent a mis en e´ vidence le critère suivant (voir propositions 3.1 et 3.2 de [Parent 2005]) : (C) Soit p ≥ 11. Supposons que pour tout j ∈ Fp non supersingulier, il existe x ∈ P0 [Ie ] tel que ι j (x) 6= 0. Alors X 0+ ( pr )(Q) est trivial c’est-àdire ne contient que des pointes et des points CM. Pour A un anneau et f : M → N un homomorphisme de Z-modules, on note M A = M ⊗ A et on note encore f : M A → N A l’homomorphisme de A-modules obtenu par extension des scalaires. Soient n le numérateur de ( p − 1)/12 et π 0 : PZ[1/n] → P0Z[1/n] la projection orthogonale (voir Section 1C). Pour D > 0 notons γ D ∈ PQ le D-ième e´ lément de Gross (voir 3A). Notons Disc( p) l’ensemble des discriminants quadratiques imaginaires1 premiers a` p. Parent déduit de la formule de Gross–Zhang que γ D0 := π 0 (γ D ) ∈ P0 [Ie ]Q pour −D ∈ Disc( p). Considérons une autre famille d’éléments du module supersingulier : (2)

ym =

g D X

Tm xi ,

i=0

E xi x ∈ P et wi i

ym0 = π 0 (ym ) ∈ P0Z[1/n]

(m ≥ 1)

où h , i est l’accouplement bilinéaire sur P défini en (3) et wi = |Aut(E i )|/2 pour E i ∈ xi . Théorème 0.1. Pour tout entier m ≥ 1, ym0 ∈ P0 [Ie ]Q . Pg Notons a E = i=0 xi /wi l’élément d’Eisenstein (voir le Section 1C). L’assertion précédente peut se déduire de la formule de Gross–Kudla ou, pour certaines valeurs de m, de la formule de Gross–Zhang et de la proposition suivante Proposition 0.2. On a ym = (m) a E +

X

γd

(m ≥ 1)

(s,d)∈Z2 4m−s 2 =dr 2 >0

où (m) = 1 si m est un carré et (m) = 0 sinon. La démonstration de la proposition 0.2 s’inspire du calcul classique qui permet d’établir la formule des traces d’Eichler (voir [Eichler 1955], [Gross 1987] et la Section 3B). Parent [2005] montre que {γ D0 , −D ∈ Disc( p)} engendre le Q-espace vectoriel P0 [Ie ]Q . Nous montrons au Section 3B les propositions suivantes : Proposition 0.3. Le Q-espace vectoriel engendré par (ym0 )m≥1 est égal au Qespace vectoriel engendré par (ym0 )1≤m≤g+1 . 1 On appelle ici discriminant quadratique imaginaire le discriminant d’un ordre d’un corps qua-

dratique imaginaire. C’est donc un carré mutiplié par un discriminant fondamental.


Proposition 0.4. Le TQ -module engendré par {ym0 , m ≥ 1} est égal à P0 [Ie ]Q . Pour√ −d < 0 un discriminant fondamental, notons εd le caractère non trivial de Gal(Q( −d)/Q). Les propositions 0.2, 0.3 et 0.4 entraˆınent une version précise d’un théorème de non-annulation : Corollaire 0.5. Si f est une forme primitive de poids 2 pour 00 ( p) telle que L( f, 1) 6= 0, alors il existe d ≤ 4g + 4 tel que L( f ⊗ εd , 1) 6= 0. Pour (m, p) = 1, ym e´ numère les boucles du graphe des m-isogénies e´ tudié par Mestre et Oesterlé [≥ 2008]. Cela permet de faire les calculs (Section 4B) conduisant au Théorème 0.6 suivant. Considérons le nombre premier p0 = 45321935159. Soit C l’ensemble des nombres premiers p qui sont un carré modulo 3, 4 et 7 et tels que l’une des conditions suivantes soit vérifiée : (1) p carré modulo 5, 11, 19, 23, 43, 67, 163, non carré modulo 8 ; (2) p carré modulo 8, 11, 19, et modulo au moins deux des nombres premiers 43, 67, 163, et vérifiant l’une des conditions suivantes (a) p carré modulo 5 ; (b) p non carré modulo 5 et 23 ; (c) p non carré modulo 5 et carré modulo 23, 59, 71 ; (d) p non carré modulo 5, 59, 71 et carré modulo 23 ; (3) p carré modulo 5, 8, 11, 43, 67, 163, non carré modulo 19 et l’une des conditions suivantes est vérifiée : (a) p carré modulo 23 ; p p p = 1. 31 36319 p0 (4) p carré modulo 5, 8, 19, 43, 67, 163, non carré modulo 11 et p carré modulo au moins un des nombres : 23, 797. (b) p non carré modulo 23 et

Théorème 0.6. Si p ≥ 11, p 6= 13 et p 6∈ C, alors X 0+ ( pr )(Q) est trivial. L’ensemble C est de densité analytique 9/210 . Parent [2005] avait obtenu un résultat analogue avec une densité de 7/29 . Le cas r = 2 de ce théorème constitue une avancée en direction du cas normalisateur d’un Cartan déployé d’un problème de Serre sur la torsion des courbes elliptiques. (Pour un e´ noncé de ce problème, se reporter a` [Serre 1972; Serre 1986, p. 288].) La liste des nombres premiers 11 ≤ p ≤ 50000, p 6= 13 dans C est donnée dans la Table 3 (page 238).

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1. Préliminaires sur le module supersingulier 1A. Réalisations géométriques et opérateurs de Hecke. Soit e T l’anneau engendré par l’action des opérateurs de Hecke Tm , m ≥ 1 sur le C-espace vectoriel M2 (00 ( p)) des formes modulaires de poids 2 pour 00 ( p). Cette action se factorise par T sur S2 (00 ( p)). Soient X 0 ( p)Z la normalisation de P1Z dans X 0 ( p) via le morphisme composé X 0 ( p) → X 0 (1) ∼ = P1Q . La fibre X 0 ( p)Fp de X 0 ( p)Z en p est constituée 1 de deux copies de PFp qui sont e´ changées par l’opérateur d’Atkin–Lehner w p et se coupent transversalement. Les points doubles de X 0 ( p)Fp sont en correspondance bijective avec les classes x0 , . . . , x g de S et g n’est autre que le genre de X 0 ( p). Notons J0 ( p) la jacobienne de X 0 ( p) et Jela jacobienne généralisée de X 0 ( p) relativement aux pointes. L’anneau e T (resp. T) est isomorphe a` l’anneau engendré par e les endomorphismes de J (resp. J0 ( p)) provenant des correspondances de Hecke sur X 0 ( p). Soient J0 ( p)Z et JeZ les modèles de Néron respectifs de J0 ( p) et Je sur Z. Le groupe P = Z[S] (resp. P0 ) s’identifie au groupe des caractères de la composante neutre de la fibre en p de JeZ (resp. J0 ( p)Z ) (voir [Raynaud 1991] et [de Shalit 1995, 2.3]). Cela définit par transport de structure une action de e T sur P qui laisse stable P0 et se factorise par T sur P0 . L’action de e T sur la classe d’isomorphisme [E] d’une courbe elliptique E supersingulière sur F¯ p est donnée P par Tm ([E]) = C [E/C] (m ≥ 1), où C parcourt l’ensemble des sous-schémas en groupes finis d’ordre m de E (voir [Raynaud 1991] ou [Mestre et Oesterlé ≥ 2008, 1.2.1]). Sur la base (xi )0≤i≤g de P, l’action de Tm (m ≥ 1) est donnée par la transposée de la matrice d’Eichler–Brandt B(m) = (Bi, j (m))0≤i, j≤g (voir [Gross 1987, sections 1 et 4]). 1B. Accouplement bilinéaire. Soit δ le dénominateur de ( p − 1)/12. Rappelons que wi = |Aut(E i )|/2 (i ∈ {0, . . . , g}, E i ∈ xi ) vérifient g Q

wi = δ

i=0

et

g P

1/wi = ( p − 1)/12.

i=0

Le Z-module P est muni de l’accouplement bilinéaire non dégénéré h , i : P × P → Z défini par (3)

hxi , x j i = wi δi, j

(0 ≤ i, j ≤ g)

où δi, j est le symbole de Krönecker. Les opérateurs de Hecke sont auto-adjoints pour h , i. Cet accouplement induit un homomorphisme injectif de e T-modules de ˇ e P dans P = Hom(P, Z) (sur lequel T agit par dualité) de conoyau isomorphe a` ˇ au sous-e Z/δ Z identifiant P T-module g L i=0

Z

xi wi


de PQ (voir [Gross 1987] ou [Emerton 2002, lemme 3.16]). L’accouplement caˇ → Z e´ tend donc l’accouplement h , i et sera encore noté h , i. nonique P × P L’homomorphisme injectif de T-modules de P0 dans Pˇ0 = Hom(P0 , Z) induit par h , i est de conoyau isomorphe a` Z/n Z (loc. cit.). Sur Pˇ0 /P0 qui s’identifie au groupe des composantes de la fibre en p de J0 ( p)Z , l’accouplement h , i n’est autre que l’accouplement de monodromie (voir [Illusie 1991] ou l’appendice de [Bertolini et Darmon 1997]). P Notons m≥0 am ( f )q m le développement de Fourier a` l’infini de f ∈ M2 (00 ( p)). Considérons le Z-module M des formes modulaires f telles que a0 ( f ) ∈ Q et am ( f ) ∈ Z pour m ≥ 1 et M0 le sous-Z-module de M constitué des formes paraT sur M2 (00 ( p)) laisse stables M et M0 et se factorise par boliques2 . L’action de e 0 T T sur M . On a MC = M2 (00 ( p)) et M0C = S2 (00 ( p)). L’accouplement sur M × e e défini par ( f, T ) 7→ a1 ( f | T ) induit un isomorphisme de T-modules (resp. de T-modules) ∼ ∼ M− → Hom(e T, Z) (resp. M0 − → Hom(T, Z)) (voir [Ribet 1983, théorème 2.2 ; Emerton 2002, proposition 1.3]). Les homomorphismes de e T-modules et T-modules ∼ ˇ e θ : P ⊗e T P → Hom(T, Z) = M P 1 m x ⊗e T y 7 → 2 (deg x. deg y) + m≥1 hTm x, yi q et θ 0 : P0 ⊗T Pˇ0 → Hom(T, Z) ∼ = M0 qui se déduisent de l’accouplement h , i, sont des surjections (voir [Emerton 2002, théorème 0.10]). e -module P. Les e 1C. Le T TQ -modules PQ et MQ et les TQ -modules P0Q et M0Q sont libres de rang 1 (voir par exemple [Gross 1987] et [Miyake 1989]). Appelons forme de Hecke une forme modulaire de poids 2 pour 00 ( p) propre pour tous les opérateurs de Hecke et normalisée, et forme primitive une forme de Hecke parabolique. On note Prim l’ensemble des formes primitives. Les idempotents primitifs de e TQ¯ sont en correspondance bijective avec les formes de Hecke et engendrent les sous-e TQ¯ -modules irréductibles de e TQ¯ . On note 1 f l’idempotent primitif associé a` une forme de Hecke f . Les opérateurs de Hecke e´ tant autoadjoints pour h , i, le e TQ¯ -module PQ¯ se décompose en somme directe de sous-espaces propres orthogof ¯ -droite naux. Pour f une forme primitive, le sous-espace PQ¯ = 1 f PQ¯ est une Q 0 dont on choisit un vecteur directeur a f . Notons σ (m) la somme des diviseurs de 2 Pour f ∈ M, on a en fait δ a ( f ) ∈ Z (voir [Emerton 2002, proposition 1.1]). Attention, les 0

notations diffèrent de celles adoptées dans [Emerton 2002] où M est noté N.

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m premiers a` p. Le sous-espace propre associé a` la série d’Eisenstein normalisée P E = ( p − 1)/24 + m≥1 σ 0 (m)q m est engendré par l’élément d’Eisenstein g X xi ˇ ⊂ PZ[1/δ] . aE = ∈P wi i=0

De plus, a E vérifie hx, a E i = deg x (x ∈ P). Par conséquent, P E := Z.a E est le ˇ → Z. On note Z-module orthogonal a` P0 pour h , i : P × P π 0 : PZ[1/n] → P0Z[1/n] 12 deg(x) a E x 7→ x − p−1 la projection orthogonale. e¯ 1D. Produits tensoriels. Considérons le e T-module P ⊗e T P. Les sous-espaces TQ g g ecrivant l’enTQ¯ -modules PQ¯ ⊗e propres de PQ¯ ⊗e ¯ sont les e TQ¯ PQ TQ¯ PQ ¯ pour g d´ semble des formes de Hecke et on a les décompositions en sous-espaces deux a` deux orthogonaux M g g E E 0 0 0 0 PQ ⊗e PQ¯ ⊗TQ¯ PQ¯ . ¯ ⊗TQ¯ PQ ¯ = TQ PQ )⊕(PQ ⊗TQ PQ ) et PQ TQ PQ = (PQ ⊗e g∈Prim g 1 f PQ¯

h En effet, pour toutes formes de Hecke f, g et h, on a ⊗e TQ¯ PQ ¯ 6 = 0 si et P g g h h seulement si f = g = h. Donc si g 6= h, on a PQ¯ ⊗e TQ¯ PQ TQ¯ PQ f 1 f PQ ¯ = ¯ = 0. ¯ ⊗e ⊗3

Considérons a` présent les e T⊗3 -modules P⊗3 , M⊗3 , et les T⊗3 -modules P0 ⊗3 et M0 . Par fonctorialité des algèbres tensorielles, on déduit de l’accouplement ˇ ⊗3 → Z. De même, le produit scalaire h , i un accouplement3 h , i⊗3 : P⊗3 × P 0 de Petersson ( , ) sur MC × MC (normalisé comme dans [Gross 1987] (7.1)) définit ⊗3

le produit scalaire de Petersson ( , )⊗3 sur M0C × MC ⊗3 (normalisé par [Gross et Kudla 1992] (11.3)). ⊗3 ⊗3 ⊗3 ⊗3 0 ⊗3 Les e TQ -modules M⊗3 et P0Q sont libres Q et PQ et les TQ -modules MQ de rang 1. Les opérateurs de Hecke triples de e T⊗3 sont autoadjoints pour ( , )⊗3 ⊗3 et h , i⊗3 . Les idempotents primitifs de e TQ¯ sont de la forme 1 F = 1 f 1 ⊗ 1 f2 ⊗ 1 f3 pour F = f 1 ⊗ f 2 ⊗ f 3 parcourant l’ensemble des formes de Hecke triples (c’est¯ -espace vectoriel a` -dire telles que f 1 , f 2 et f 3 soient des formes de Hecke). Le Q ⊗3 F ¯ PQ¯ = 1 F PQ¯ est une Q-droite de vecteur directeur A F = a f 1 ⊗Q¯ a f2 ⊗Q¯ a f3 . On a les décompositions en sous-espaces propres deux a` deux orthogonaux ⊗3 M M F F P⊗3 = P et P0Q¯ = PQ ¯ ¯, ¯ Q Q F 3 donné par ha ⊗ a ⊗ a , b ⊗ b ⊗ b i⊗3 = Q3 ha , b i 1 2 3 1 2 3 i=1 i i

F


la somme directe portant respectivement sur l’ensemble des formes de Hecke triples et l’ensemble des formes primitives triples. On vérifie aisément que pour toute forme de Hecke triple F, on a 1F X =

(4)

hA F , X i⊗3 AF hA F , A F i⊗3

⊗3 (X ∈ PQ ¯ ).

2. Formule de Gross–Kudla et éléments de P0 [Ie ] Soit

g X 1 ⊗3 x ∈ P⊗3 13 = Q wi i i=0

l’élément diagonal de Gross–Kudla. Posons s : P ⊗ P P ⊗e T P la surjection canonique, ¯ 3 = (1 ⊗Q s)(13 ) ∈ PQ ⊗Q PQ ⊗e 1 TQ PQ et ¯ 03 = (π 0 ⊗Q s)(13 ) = (π 0 ⊗Q 1)(1 ¯ 3 ) ∈ P0Q ⊗Q PQ ⊗e 1 TQ PQ . Gross et Kudla [1992, théorème 11.1] ont montré que pour F une forme primitive triple, on a (F, F)⊗3 L(F, 2) = h1 F 13 , 1 F 13 i⊗3 . 4π p En particulier, lorsque F = f ⊗ h ⊗ h ( f, h ∈ Prim), par [Gross et Kudla 1992, (11.7)], on obtient (5)

L( f, 1)L( f ⊗ Sym2 h, 2) =

(F, F)⊗3 h1 F 13 , 1 F 13 i⊗3 . 4π p

Nous allons voir que cela entraˆıne le Théorème 2.1. On a ¯ 03 1

12 ¯3− =1 a E ⊗Q p−1

et

X g

xi ⊗e TQ

i=0

xi wi

¯ 03 ∈ P0 [Ie ]Q ⊗ (P0Q ⊗TQ P0Q ). 1 Remarque 2.2. Le T-module P0 est localement libre après extension des scalaires a` Z[1/2] (voir l’introduction de [Emerton 2002]), par conséquent il y a au pire de la 2-torsion dans P0 [Ie ] ⊗ (P0 ⊗T P0 ). Ainsi, puisque δ13 ∈ P⊗3 , on a même ¯ 0 ∈ P0 [Ie ] ⊗ (P0 ⊗T P0 ) a` un e´ lément de 2-torsion près. nδ 1 3 Démonstration. On a ¯ 03 = 1

g X i=0

xi −

xi 12 a E ⊗Q xi ⊗e , TQ p−1 wi

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MARUSIA REBOLLEDO

ce qui prouve la première assertion du Théorème 2.1. On a dans P0Q¯ ⊗Q¯ (PQ¯ ⊗e ¯) TQ¯ PQ X X ¯ 03 = ¯ 3) + ¯ 3) 1 (1 f ⊗ 1h )(1 (1 f ⊗ 1 E )(1 f,h∈Prim

f ∈Prim

X X (1 f ⊗ 1h ⊗ 1h )(13 ) + (1 f ⊗ 1 E ⊗ 1 E )(13 ) . = (1 ⊗ s) f,h∈Prim

f ∈Prim

Pour tout f ∈ Prim, on a (voir (4)) h13 , a f ⊗ a E ⊗ a E i⊗3 (a f ⊗ a E ⊗ a E ). ha f ⊗ a E ⊗ a E , a f ⊗ a E ⊗ a E i⊗3 Pg Or h13 , a f ⊗ a E ⊗ a E i⊗3 = i=0 w1i hxi , a f i hxi , a E i2 = ha E , a f i = 0. Par conséquent, X 0 ¯ (1 f ⊗ 1h ⊗ 1h )(13 ) 13 = (1 ⊗ s) (1 f ⊗ 1 E ⊗ 1 E )(13 ) =

f,h∈Prim

X X = (1 ⊗ s) (1 f ⊗ 1h ⊗ 1h )(13 ) + (1 f ⊗ 1h ⊗ 1h )(13 ) . f,h∈Prim L( f,1)6 =0

f,h∈Prim L( f,1)=0

Puisque ( , )⊗3 et h , i⊗3 sont définis positifs, on déduit de (5) que lorsque f ∈ Prim est telle que L( f, 1) = 0, alors (1 f ⊗ 1h ⊗ 1h )(13 ) = 0 (h ∈ Prim). Par ailleurs, rappelons que l’idéal Ie de T est l’annulateur de l’ensemble des formes paraboliques f pour lesquelles L( f, 1) 6= 0 (voir [Merel 1996]). On en déduit que, lorsque f ∈ Prim est telle que L( f, 1) 6= 0, le vecteur propre associé a f est dans P0Q¯ [Ie ] et donc (1 f ⊗ 1h ⊗ 1h )(13 ) ∈ P0 [Ie ]Q ⊗Q P0Q ⊗Q P0Q (h ∈ Prim). On déduit immédiatement du Théorème 2.1 le corollaire suivant : ¯ 0) ∈ Corollaire 2.3. Pour toute forme Q-linéaire φ sur P0Q ⊗TQ P0Q , on a (1⊗Q φ)(1 3 P0 [Ie ]Q . Par définition de θ, on a (6)

¯ 3 ) et ym = (1 ⊗ (am ◦ θ ))(1

¯ 03 ). ym0 = (1 ⊗Q (am ◦ θ 0 ))(1

Le Théorème 0.1 e´ noncé dans l’introduction se déduit alors du corollaire 2.3 appliqué a` φ = am ◦ θ 0 . Remarque 2.4. Toute forme linéaire sur P0Q ⊗TQ P0Q s’obtient comme combinaison linéaire des formes linéaires am ◦ θ 0 (m ≥ 1) car θ 0 : PQ ⊗TQ P0Q → M0Q est un isomorphisme. Il n’est donc pas restrictif de considérer les combinaisons linéaires d’éléments ym0 comme nous le faisons par la suite.


3. Les éléments ym 3A. Comparaison avec les éléments de Gross. Soit D > 0. Notons O−D l’ordre quadratique de discriminant −D s’il existe et u(−D) l’ordre de O∗−D /h±1i. Pour i ∈ {0, . . . , g}, les anneaux Ri = End E i , i ∈ {0, . . . , g}, sont des ordres maximaux de l’algèbre de quaternions B sur Q ramifiée en p et ∞ ; on note h i (−D) le nombre de plongements optimaux de O−D dans Ri modulo conjugaison par Ri∗ . Le D-ième e´ lément de Gross4 est défini par g

(7)

X 1 γD = h i (−D)xi ∈ PZ[1/6] . 2u(−D) i=0

On a γ D = 0 si −D n’est √ pas un discriminant quadratique imaginaire ou bien si p est décomposé dans Q( −D) (en effet, dans ce dernier cas, O−D ne se plonge pas dans l’algèbre de quaternions ramifiée en p et l’infini). Démonstration de la proposition 0.2. Soit m ≥ 1 un entier. L’opérateur Tm agissant Pg sur xi par la transposée de la matrice de Brandt B(m), on a ym = i=0 Bi,i (m)xi . Notons N la norme réduite et tr la trace sur B. On a 1 1 X Bi,i (m) = Card{b ∈ Ri ; N (b) = m} = Card(Ai (s, m)) 2wi 2wi s∈Z s 2 ≤4m

où Ai (s, m) = {b ∈ Ri ; N (b) = m, tr(b) = s}. Posons D = 4m − s 2 . Lorsque D = 0, ce qui est possible si et seulement si m est un carré, Ai (s, m) n’a qu’un seul e´ lément. Considérons maintenant le cas où D > 0. Les e´ léments de Ai (s, m) sont en bijection avec les plongements de O−D dans Ri . Pour chaque tel plongement f il existe un unique ordre O−d contenant O−D (D = dr 2 pour un certain r ) tel que f s’étende en un plongement optimal de O−d dans Ri . On a donc une partition G Ai (s, m) = Ai (s, m)d d∈N;∃r ∈N; dr 2 =D

où les e´ léments de Ai (s, m)d ⊂ Ai (s, m) correspondent aux plongements de O−D dans Ri qui s’étendent en un plongement optimal de O−d dans Ri . Comme Card(Ai (s, m)d ) = h i (−d) | Ri× /O× −d |= wi h i (−d)/u(−d), 4 Cet élément, introduit par Gross, est noté e

D dans [Gross 1987] et [Parent 2005].

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X h i (−d) on a finalement Card(Ai (s, m)) = wi . Par conséquent, pour tout u(−d) entier m > 0, on a d∈N;∃r ∈N dr 2 =D

ym =

X

g X

X

(s,d)∈Z×N ∃r >0; 4m−s 2 =dr 2 >0 g X X h i (−d) aE + xi . 2u(−d) i=0 (s,d)∈Z×N 4m−s 2 =dr 2 >0

4m=s 2 i=0

= (m)

xi + 2wi

g X i=0

h i (−d) xi 2u(−d)

Remarque 3.1. Le raisonnement précédent est celui qui donne la formule d’Eichler pour la trace de Tm (voir [Eichler 1955] ou [Gross 1987]). On retrouve cette formule en identifiant les degrés de chacun des membres de l’égalité de la proposition 0.2. Remarque 3.2. La formule de Gross [1987, corollaire 11.6] montre que pour tout entier d < 0 premier a` p, l’élément γd0 est dans P0 [Ie ]Q (voir par exemple [Parent 2005]). La proposition 0.2 donne alors une nouvelle preuve du Théorème 0.1 dans le cas particulier où m est tel que tout entier d > 0 tel que 4m − s 2 = dr 2 soit premier a` p. ` titre d’exemple, voici la décomposition de ym comme combinaison linéaire A d’éléments de Gross pour m ≤ 13 : y1 = a E + 2γ3 + γ4 , y2 = 2γ4 + 2γ7 + γ8 , y3 = 3γ3 + 2γ8 + 2γ11 + γ12 , y4 = a E + 2γ3 + γ4 + 2γ7 + 2γ12 + 2γ15 + γ16 , y5 = 4γ4 + 2γ11 + 2γ16 + 2γ19 + γ20 , y6 = 2γ8 + 2γ15 + 2γ20 + 2γ23 + γ24 , y7 = 6γ3 + γ7 + 2γ12 + 2γ19 + 2γ24 + 2γ27 + γ28 , y8 = 2γ4 + 4γ7 + γ8 + 2γ16 + 2γ23 + 2γ28 + 2γ31 + γ32 , y9 = a E + 2γ3 + γ4 + 2γ8 + 2γ11 + 2γ20 + 2γ27 + 2γ32 + 2γ35 + γ36 , y10 = 4γ4 + 2γ15 + 2γ24 + 2γ31 + 2γ36 + 2γ39 + γ40 , y11 = 2γ7 + 2γ8 + γ11 + 2γ19 + 2γ28 + 2γ35 + 2γ40 + 2γ43 + γ44 , y12 = 3γ3 + 2γ8 + 2γ11 + 3γ12 + 2γ23 + 2γ32 + 2γ39 + 2γ44 + 2γ47 + γ48 , y13 = 6γ3 + 4γ4 + 2γ12 + 2γ16 + 2γ27 + 2γ36 + 2γ43 + 2γ48 + 2γ51 + γ52 . 3B. Espace vectoriel et module de Hecke engendrés par ces éléments. Nous démontrons dans ce paragraphe les propositions 0.3 et 0.4 ainsi que le corollaire


0.5. Faisons tout d’abord quelques observations. A une forme linéaire φ sur PQ , ¯ 0 ) ∈ M0 . Remarquons que gφ a associons la forme modulaire gφ = (φ ⊗Q θ 0 )(1 3 Q pour q-développement X (8) φ(ym0 )q m . m≥1

De plus, puisque φ ⊗Q¯ θ 0 est un homomorphisme de TQ¯ ⊗Q¯ TQ¯ -modules, on a X 0 ¯ (9) 1h gφ = (φ ⊗Q¯ θ ) (1g ⊗Q¯ 1h )(13 ) (h ∈ Prim). g∈Prim

Démonstration de la proposition 0.3. Il suffit de montrer que toute forme linéaire 0 est sur l’espace vectoriel engendré par (ym0 )m≥1 et qui s’annule en y10 , . . . , yg+1 nulle. Soit φ une telle forme linéaire. La forme différentielle ωφ de X 0 ( p) associée a` gφ a pour q-développement X dq φ(ym0 )q m . q m≥1

0 Si φ(y10 ) = 0, . . . , φ(yg+1 ) = 0, la forme différentielle holomorphe ωφ a un zéro d’ordre g en l’infini. L’infini n’étant pas un point de Weierstrass de X 0 ( p) (voir [Ogg 1978]), on en déduit que ωφ est nulle, d’où la proposition.

Pour démontrer la proposition 0.4 on a encore besoin de deux lemmes : Lemme 3.3. Soient f et h deux formes primitives. Les assertions suivantes sont équivalentes : a) L( f, 1)L( f ⊗ Sym2 h, 2) = 0 ;

b) 1h .gh .,a f i = 0.

Démonstration. D’après (9), on a 1h gh . ,a f i = h . , a f i ⊗Q¯ θ

0

X

¯ 3) . (1g ⊗Q¯ 1h )(1

g∈Prim

Puisque hag , a f i = 0 si g 6= f , on obtient ¯ 3 ). 1h gh.,a f i = (h . , a f i ⊗Q¯ θ 0 )(1 f ⊗Q¯ 1h )(1 Or l’application f

h . , a f i ⊗Q¯ θ 0 : PQ¯ ⊗Q¯ (P0Q¯ ⊗TQ¯ P0Q¯ ) → M0Q¯ est injective car θ 0 est un isomorphisme de TQ -modules libres (voir fin du Section ¯ 3 ) = 0 i.e. 1A). Par conséquent 1h gh . ,a f i = 0 si et seulement si (1 f ⊗Q¯ 1h )(1 si et seulement si (1 f ⊗Q¯ 1h ⊗Q¯ 1h )(13 ) = 0. D’après (5), ceci est e´ quivalent a` l’assertion a).

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Lemme 3.4. Les conditions suivantes sont équivalentes : i) le TQ -module Y engendré par {ym0 , m ≥ 1} est égal à P0 [Ie ]Q ; ii) pour toute forme primitive f de poids 2 pour 00 ( p) telle que L( f, 1) 6= 0, il existe une forme primitive h de poids 2 pour 00 ( p) telle que L( f ⊗ Sym2 h, 2) 6= 0. ¯ -droites, on a Y = Démonstration. Les espaces TQ¯ -propres de P0Q¯ e´ tant des Q P0 [Ie ]Q si et seulement si pour tout f ∈ Prim tel que L( f, 1) 6 = 0, il existe m ≥ 1 tel que 1 f ym 6= 0. Par ailleurs, d’après le lemme 3.3, la condition ii) du lemme est vérifiée si et seulement si pour tout f ∈ Prim tel que L( f, 1) 6= 0, on a gh . ,a f i 6= 0 (car il existe alors h telle que 1h gh . ,a f i 6= 0). Puisque gh . ,a f i a pour q-développement P m eduit l’équivalence des assertions i) et ii). m≥1 hym , a f i q (voir (8)), on en d´ Démonstration de la proposition 0.4. D’après les travaux de Böcherer et SchulzePillot [1999] améliorés par Arakawa et Böcherer [2003, théorème 5.3], la condition ii) du lemme ci-dessus est toujours satisfaite. Démonstration du corollaire 0.5. Soit f ∈ Prim telle que L( f, 1) 6= 0. D’après les propositions 0.3 et 0.4, il existe alors 1 ≤ m ≤ g + 1 tel que 1 f ym 6= 0. Par conséquent, il existe D ≤ 4g +4 tel que 1 f γ√D 6= 0. On a (D, p) = 1 car D ≤ 4g +4. Soit −d le discriminant fondamental de Q( −D). En vertu des relations de norme [Bertolini et Darmon 1996] 2.4, on a γ D ∈ e T.γd . On en déduit que 1 f γd 6 = 0 et L( f ⊗ εd , 1) 6= 0 d’après la formule de Gross [1987, corollaire 11.6] généralisée par Zhang [2001, théorème 1.3.2]. 4. Points rationnels de courbes modulaires 4A. Une question théorique. On suppose désormais p ≥ 11. Soit Z( p) le localisé de Z en p. Soit j ∈ P1 (F¯ p ) non supersingulier. On e´ tend ι j a` PZ( p) en posant ι j (1/a) = a −1 . Supposons vérifiée la condition : (H)

Pour tout j ∈ Fp ordinaire, il existe m ≥ 1 tel que ι j (ym0 ) 6= 0.

D’après le Théorème 0.1 et le critère (C) e´ noncé dans l’introduction, on a alors p assez grand (par exemple p > 37), l’hypothèse (H) est vérifiée. Dans l’espoir d’obtenir un résultat sans les contraintes de congruence du Théorème 0.6, remarquons que l’on peut reformuler 0 ¯ 0 ) ∈ P0 cette condition. On a (1⊗θ 0 )(1 erons la forme parabolique 3 Z[1/n] ⊗M . Consid´ 0 a` coefficients dans F¯ p (i.e. l’élément de F¯ p ⊗ M ) définie par X 0+ ( pr )(Q) trivial. Actuellement, on ne sait pas si pour

¯ 03 ) ∈ M0¯ . g j = (ι j ⊗ θ 0 )(1 Fp P Cette forme modulaire a pour q-développement m≥1 ι j (ym0 ) q m . L’hypothèse (H) est donc e´ quivalente a` : (10)


(H0 )

Pour tout j ∈ Fp ordinaire, on a g j 6= 0.

4B. Calculs pratiques : démonstration du Théorème 0.6. Soit j ∈ Fp non supersingulier. On a ι j (ym0 ) =

g g X X 1 Bi,i (m) + 12 tr(B(m)) ∈ F¯ p . j − ji wi ( j − ji ) i=0

i=0

En effet, ym0

12 deg(ym )a E = ym − p−1

et ι j (a E ) =

g X i=0

1 . wi ( j − ji )

Le membre de droite de cette dernière e´ galité n’étant pas facile a` calculer, nous introduisons yk,m = tr B(m)yk − tr B(k)ym ∈ P0 [Ie ]Q

(0 < k < m)

et calculons ι j (yk,m ) pour k, m dans {2, 3, 5, 6, 7}. Lorsque (m, p) = 1, l’entier Bi,i (m) est la multiplicité de ji comme racine du polynôme modulaire φm (X, X ) dans F¯ p (voir [Igusa 1959] ou [Lang 1987, 5.3 théorème 5]). Les polynômes modulaires φm (X, X ) pour m ∈ {2, 3, 5, 6, 7}, donnés par Magma, sont : φ2 (X, X ) = −(X − 1 728)(X + 3 375)2 (X − 8 000), φ3 (X, X ) = −X (X − 54 000)(X + 32 768)2 (X − 8 000)2 , φ5 (X, X ) = −(X − 1 728)2 (X − 287 496)2 (X + 32 768)2 (X + 884 736)2 P1 , , φ6 (X, X ) = (X − 8 000)2 P12 P2 P32 P42 , φ7 (X, X ) = −(X + 3 375)(X − 16 581 375)X 2 (X − 54 000)2 × (X + 12 288 000)2 (X + 884 736)2 P22 , où P1 = X 2 − 1 264 000 X − 681 472 000, P2 = X 2 − 4 834 944 X + 14 670 139 392, P3 = X 2 + 191 025 X − 121 287 375, P4 = X 3 + 3 491 750 X 2 − 5 151 296 875 X + 12 771 880 859 375. Comme m n’est pas carré, les racines de φm (X, X ) dans C sont les invariants j (τ ) de courbes elliptiques a` multiplication complexe par un ordre quadratique imaginaire Z[τ ]. Les ordres quadratiques associés aux invariants racines des facteurs de degré 1 sont donnés dans la littérature (voir par exemple [Cohen 1993, 7.2.3]). En

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déterminant les ordres quadratiques imaginaires possédant un e´ lément de norme 5, 6 et 7, on trouve une racine αi du facteur Pi de degré 2 pour i ∈ {1, . . . , 4}. On obtient ainsi les valeurs de Bi,i (m) (m ∈ {2, 3, 5, 6, 7}) données dans la table ci-dessous où a, b, c, d, e, f, g, h, v, w ∈ {0, 1} sont par définition e´ gaux a` 1 si et seulement si j (τ ) est supersingulier modulo p, c’est-à-dire si p est inerte ou ramifié dans Q(τ ). On supposera désormais que p > 173. Dans ce cas, les invariants apparaissant dans cette table sont tous distincts. L’égalité a = c e´ quivaut a` h = d et a = d e´ quivaut a` v = g. On fait parcourir au 8-uplet (a, b, c, d, e, f, g, w) les différentes valeurs possibles. Pour tous les 8-uplets distincts de ceux e´ numérés dans la Table 2, les fractions ι j (yk,m ), où k, m ∈ {2, 3, 5, 6, 7}, ne s’annulent pas simultanément. La table résume les résultats obtenus pour tous les 8-uplets posant un problème. Lorsque ι j (yk,m ) n’est pas une fraction identiquement nulle, on note n k,m le degré de son numérateur. Lorsque n k,m = 2, ι j (yk,m ) a un zéro dans Fp si et seulement si le discriminant dk,m de son numérateur (dans Z) est un carré modulo p.

ji 1728 287496 − 3375 16581375 8000 0 54000 − 12288000 − 32768 − 884736 α1 α2 α3 α4

τ √ −1 √ 2 −1 √ 1 2 (1 + −7) √ −7 √ −2 √ 1 2 (1 + −3) √ −3 √ 1 2 (1 + 3 −3) √ 1 (1 + −11) 2 √ 1 2 (1 + −19) √ −5 √ −6 √ 1 2 (1 + −15) √ 1 2 (1 + −23)

Bi (2) Bi (3) Bi (5) Bi (6) Bi (7)

Dτ 4

a

1

2

4

a

7

b

7

b

8

c

3

d

1

2

3

d

1

2

3

d

11

e

19

f

2

20

g

1

24

h

1

15

v

2

23 w

2

2 2

1 1

1

2

2

2 2

2 2 2

Table 1. Valeurs de Bi (m) = Bi,i (m), m ∈ {2, 3, 5, 6, 7}.

2


a b c d

e f

g w

Résultat

0 0 1 0

0 0

0 0

voir texte ci-dessous

0 0 0 0

0 0

0 ∗ 1 0

ι j (yk,m ) = 0 (k, m ∈ {2, 3, 5, 6, 7}) n 5,6 = 2, d5,6 = 26 .56 .115 .134 .372 .59.71, ι j (yk,m ) = 0 (k, m) 6= (5, 6) n 5,6 = 5, ι j (yk,m ) = 0 (k, m) 6= (5, 6)

1 1 0 1

0 0 0 1

ι j (yk,m ) = 0 (k, m ∈ {2, 3, 5, 6, 7}) ι j (y5,6 ) = ι j (y6,7 ) n 5,6 = 2, d5,6 = 26 .57 .74 .31.36319. p0 ι j (yk,m ) = 0 si (k, m) 6= (5, 6), (6, 7)

1 0

0 0 0 1

ι j (yk,m ) = 0 (k, m ∈ {2, 3, 5, 6, 7}) ι j (y3,6 ) = ι j (y5,6 ), n 5,6 = 2, d5,6 = 26 .3.57 .78 .112 .172 .192 .797 ι j (yk,m ) = 0 si (k, m) 6= (3, 6), (5, 6)

Table 2. Résultats des calculs pous les cas exceptionnels. Le symbole ∗ signifie que le coefficient peut prendre indifféremment la valeur 0 ou 1. On rappelle que p0 = 45321935159. ` titre d’exemple, lorsque (a, b, c, d, e, f, g, w) = (0, 0, 1, 0, 0, 0, 0, 0), on obA tient ι j (y2,3 ) = ι j (y2,5 ) = ι j (y3,5 ) = ι j (y5,6 ) = ι j (y5,7 ) = 0 et ι j (y2,6 ), ι j (y3,6 ), ι j (y2,7 ), ι j (y3,7 ) et ι j (y6,7 ) sont, a` multiplication par une puissance de 2 près, e´ gaux a` Q( j) =

P 0 ( j) −211 (13.181 j + 26 .33 .7.13.29) 2 − 2 = j − 8000 P2 ( j) ( j − 8000)P2 ( j)

qui s’annule en j0 ≡ −(181)−1 .26 .33 .7.29 mod p. Posons B l’ensemble des nombres premiers p qui sont simultanément un carré modulo 3, 4 et 7 et qui vérifient l’une des conditions suivantes : i) p carré modulo 5, 11, 19, et 23 et non carré modulo 8, ii) p est un carré modulo 5, 8, 11 et 19 ; iii) p carré modulo 8, 11 et 19, non carré modulo 5, 23 ; iv) p carré modulo 8, 11, 19, 23, 59, 71, non carré modulo 5 ; v) p carré modulo 8, 11, 19, 23, non carré modulo 5, 59, 71 ;

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MARUSIA REBOLLEDO

vi) p carré modulo 5, 8, 11, 23, non carré modulo 19 ;

vii) p carré modulo 5, 8, 11, non carré modulo 19, 23 et

p 31

p 36319

viii) p carré modulo 5, 8, 19, 23, non carré modulo 11 ;

p = 1; p0

ix) p carré modulo 5, 8, 19, 797, non carré modulo 11, 23. Lemme 4.1. Si p > 173, p 6= 797, 36319, p0 et p 6∈ B, alors pour tout j ∈ Fp non supersingulier, il existe (k, m) ∈ {2, 3, 5, 6, 7}2 tel que ι j (yk,m ) 6= 0 et par conséquent X 0+ ( pr )(Q) est trivial. Démonstration. On vérifie aisément que lorsque p est comme dans l’énoncé et j ∈ Fp , nous ne sommes pas dans l’un des cas e´ numérés dans la Table 2 et donc il y a une fraction non nulle parmi ι j (yk,m ), (k, m) ∈ {2, 3, 5, 6, 7}2 . Le lemme 4.1 et le théorème 1.1 de [Parent 2005] entraˆınent alors le Théorème 0.6. En effet, l’ensemble C du Théorème 0.6 est e´ gal a` A ∩ B et 797, 36319, p0 ne sont pas dans A (ici A est l’ensemble du théorème 1.1 de [Parent 2005]). Le cas des nombres premiers p = 11 et 17 ≤ p ≤ 173 a e´ té traité par Parent [2005, p. 8, preuve du théorème 1.1]. 1873 3217 7417 8233 9241 10333 11257 15733 16921 17389 18313 19273 21961 26161 26497 26833 30097 31081 32377 34057 35281 36793 38329 38833 41617 42337 42793 48409 Table 3. Nombres premiers p ≤ 50000 dans l’ensemble C.

Remerciements Je tiens ici a` remercier L. Merel, P. Parent ainsi que le referee pour leurs remarques durant l’élaboration de cet article. References [Arakawa et Böcherer 2003] T. Arakawa et S. Böcherer, “Vanishing of certain spaces of elliptic modular forms and some applications”, J. Reine Angew. Math. 559 (2003), 25–51. MR 2004m:11069 Zbl 1043.11040 [Bertolini et Darmon 1996] M. Bertolini et H. Darmon, “Heegner points on Mumford–Tate curves”, Invent. Math. 126:3 (1996), 413–456. MR 97k:11100 Zbl 0882.11034 [Bertolini et Darmon 1997] M. Bertolini et H. Darmon, “A rigid analytic Gross–Zagier formula and arithmetic applications”, Ann. of Math. (2) 146:1 (1997), 111–147. MR 99f:11079 Zbl 1029.11027 [Böcherer et Schulze-Pillot 1999] S. Böcherer et R. Schulze-Pillot, “Squares of automorphic forms on quaternion algebras and central values of L-functions of modular forms”, J. Number Theory 76:2 (1999), 194–205. MR 2000h:11046 Zbl 0940.11023


[Cohen 1993] H. Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics 138, Springer, Berlin, 1993. MR 94i:11105 Zbl 0786.11071 [Eichler 1955] M. Eichler, “Zur Zahlentheorie der Quaternionen-Algebren”, J. Reine Angew. Math. 195 (1955), 127–151. MR 18,297c Zbl 0068.03303 [Emerton 2002] M. Emerton, “Supersingular elliptic curves, theta series and weight two modular forms”, J. Amer. Math. Soc. 15:3 (2002), 671–714. MR 2003b:11038 Zbl 01739913 [Gross 1987] B. H. Gross, “Heights and the special values of L-series”, pp. 115–187 dans Number theory (Montreal, 1985), édité par H. Kisilevsky et J. Labute, CMS Conf. Proc. 7, Amer. Math. Soc., Providence, RI, 1987. MR 89c:11082 Zbl 0623.10019 [Gross et Kudla 1992] B. H. Gross et S. S. Kudla, “Heights and the central critical values of triple product L-functions”, Compositio Math. 81:2 (1992), 143–209. MR 93g:11047 Zbl 0807.11027 [Igusa 1959] J.-i. Igusa, “Kroneckerian model of fields of elliptic modular functions”, Amer. J. Math. 81 (1959), 561–577. MR 21 #7214 Zbl 0093.04502 [Illusie 1991] L. Illusie, “Réalisation l-adique de l’accouplement de monodromie d’après A. Grothendieck”, 196–197 (1991), 27–44. MR 93c:14020 Zbl 0781.14011 [Lang 1987] S. Lang, Elliptic functions, 2e éd., Graduate Texts in Mathematics 112, Springer, New York, 1987. MR 88c:11028 Zbl 0615.14018 [Merel 1996] L. Merel, “Bornes pour la torsion des courbes elliptiques sur les corps de nombres”, Invent. Math. 124:1-3 (1996), 437–449. MR 96i:11057 Zbl 0936.11037 [Mestre et Oesterlé ≥ 2008] J.-F. Mestre et J. Oesterlé, “Courbes elliptiques de conducteur premier”. Please supply year Non publié. [Miyake 1989] T. Miyake, Modular forms, Springer, Berlin, 1989. MR 90m:11062 Zbl 0701.11014 [Momose 1984] F. Momose, “Rational points on the modular curves X split ( p)”, Compositio Math. 52:1 (1984), 115–137. MR 86j:11064 Zbl 0574.14023 [Momose 1986] F. Momose, “Rational points on the modular curves X 0+ ( pr )”, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 33:3 (1986), 441–466. MR 88a:11056 Zbl 0621.14018 [Momose 1987] F. Momose, “Rational points on the modular curves X 0+ (N )”, J. Math. Soc. Japan 39:2 (1987), 269–286. MR 88h:14031 Zbl 0623.14009 [Ogg 1978] A. P. Ogg, “On the Weierstrass points of X 0 (N )”, Illinois J. Math. 22:1 (1978), 31–35. MR 57 #3136 Zbl 0374.14005 [Parent 2005] P. J. R. Parent, “Towards the triviality of X 0+ ( pr )(Q) for r > 1”, Compos. Math. 141:3 (2005), 561–572. MR 2006a:11076 Zbl 02183028 [Raynaud 1991] M. Raynaud, “Jacobienne des courbes modulaires et opérateurs de Hecke”, 196-197 (1991), 9–25. MR 93b:11077 Zbl 0781.14020 [Ribet 1983] K. A. Ribet, “Mod p Hecke operators and congruences between modular forms”, Invent. Math. 71:1 (1983), 193–205. MR 84j:10040 Zbl 0508.10018 [Serre 1972] J.-P. Serre, “Propriétés galoisiennes des points d’ordre fini des courbes elliptiques”, Invent. Math. 15:4 (1972), 259–331. Réimpression: pp. 1–73 dans ses Œuvres, tome 3, Springer, Berlin, 1986. MR 52 #8126 Zbl 0235.14012 [Serre 1986] J.-P. Serre, “Resume des cours 1975–1976”, pp. 284–291 dans ses Œuvres, vol. III, Springer, Berlin, 1986. MR 89h:01109c [de Shalit 1995] E. de Shalit, “On certain Galois representations related to the modular curve X 1 ( p)”, Compositio Math. 95:1 (1995), 69–100. MR 96i:11063 Zbl 0853.11045 [Zhang 2001] S.-W. Zhang, “Gross–Zagier formula for GL2 ”, Asian J. Math. 5:2 (2001), 183–290. MR 2003k:11101 Zbl 01818531

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Received December 1, 2006. Revised October 15, 2007. M ARUSIA R EBOLLEDO L ABORATOIRE DE M ATH E´ MATIQUES U NIVERSIT E´ B LAISE PASCAL C AMPUS UNIVERSITAIRE DES C E´ ZEAUX 63177 AUBI E` RE F RANCE [email protected] http://math.univ-bpclermont.fr/~rebolledo/


TWO REMARKS ON A THEOREM OF DIPENDRA PRASAD H IROSHI S AITO We show two results on local theta correspondence and restrictions of irreducible admissible representations of GL(2) over p-adic fields. Let F be a nonarchimedean local field of characteristic 0, and let L be a quadratic extension of F. Let L/F is the character of F × corresponding to the extension L/F, and let GL2 (F)+ be the subgroup of GL2 (F) consisting of elements with L/F (det g) = 1. The first result is that the theorem of Moen–Rogawski on the theta correspondence for the dual pair (U(1), U(1)) is equivalent to a result by D. Prasad on the restriction to GL2 (F)+ of the principal series representation of GL2 (F) associated with 1, L/F . As the second result, we show that we can deduce from this a theorem of D. Prasad on the restrictions to GL2 (F)+ of irreducible supercuspidal representations of GL2 (F) associated to characters of L × .

1. Introduction The purpose of this paper is to give two remarks on the comment in the last Remark in Section 3 of [Prasad 2007] and Theorem 1.2 in [Prasad 1994]. Let F be a nonarchimedean local field of characteristic 0, and let L be an quadratic extension of F. We denote by L/F the quadratic character of F × corresponding to the extension L/F. Let Ps(1, L/F ) be the normalized principal series representation of GL2 (F) associated to the characters 1 and L/F . We fix an embedding of L × into GL2 (F). The restriction of Ps(1, L/F ) to L × is a multiplicity-free direct sum. Let GL2 (F)+ be the subgroup of GL2 (F) consisting of elements with determinant belonging to N L/F (L × ). Then L × is contained in GL2 (F)+ , and the restriction of Ps(1, L/F ) to GL2 (F)+ decomposes into two irreducible subspaces Ps± (1, L/F ). In this situation, Lemma 4 in [Prasad 2007] states that a character φ of L × , whose restriction to F × is L/F , appears in Ps+ (1, L/F ) (resp. Ps− (1, L/F )) if and only if ε(φ, ψ0 ) = 1 (resp. −1). Here ψ0 is a character of L, the precise definition of which will be given in Section 3. On the other hand, we fix a character χ of L × whose restriction to MSC2000: primary 22E50, 11F27; secondary 11F70. Keywords: theta correspondence, epsilon factor. 241

242

HIROSHI SAITO

F × is L/F , and consider the theta correspondence for the dual pair (U (1), U (1)) with respect to χ . Then the theorem of Moen–Rogawski states that a character η of L 1 appears in this theta correspondence if and only if ε(χη−1 L , ψ0 ) = 1 (see × [Moen 1987; Rogawski 1992]). Here η L is the character of L given by η L (x) = η(x/x) ¯ for x ∈ L × . Now the correspondence η 7→ χη−1 L yields a one to one correspondence 1 between characters of L and characters of L × whose restriction to F × is L/F . Thus the factor ε(φ, ψ0 ) appears in formulas expressing characters of linear and nonlinear groups. The Remark in Section 3 of [Prasad 2007] raises the question whether there is a natural explanation for this phenomenon. Our first remark is an answer to this question. Our result is that Lemma 4 in Prasad’s article is equivalent to the theorem of Moen–Rogawski. We show this in Sections 3 and 4 using seesaw diagrams after some preparations on seesaw diagrams in Section 2. We note that both the theorem of Moen–Rogawski and Prasad’s Lemma 4 were originally proved by local methods for F with odd residual characteristic, and the general cases were proved by these local results and global methods (see [Moen 1987], Proposition 3.4 of [Rogawski 1992], and Lemma 4 of [Prasad 2007]). Later a purely local proof for the theorem of Moen–Rogawski was given by Harris, Kudla and Sweet (see Corollaries 8.5 and A.9 of [Harris et al. 1996]), and that of Lemma 4 of [Prasad 2007] was given by the author (see Appendix of [Prasad 2007]). The second remark is concerned with Theorem 1.2 in [Prasad 1994]. Let π be the irreducible supercuspidal representation of GL2 (F) associated to a character λ of L × by theta correspondence. Then π| L × is multiplicity-free, and π|GL2 (F)+ decomposes into two irreducible subspaces π + and π − . In the article in question, D. Prasad proved that φ with λφ −1 | F × = L/F appears in π ± if and only if ¯ −1 , ψ0 ) = ±1. In Section 3 we deduce an analogue of this ε(λφ −1 , ψ0 ) = ε(λφ theorem for unitary groups of degree 2 (Theorem 3.5) from the theorem of Moen– Rogawski using a seesaw diagram. In Section 4 we show the above theorem of D. Prasad from this again using a seesaw diagram, which is found in [Harris 1993]. This is the first half of Theorem 1.2 in [Prasad 1994]. In Section 5, we treat a similar problem for representations of multiplicative group of the division quaternion algebra. This is the second half of Theorem 1.2 in [Prasad 1994]. 2. Seesaw diagrams In this section, we introduce notation and recall some seesaw diagrams which will be used in later sections. Let F, L and L/F be as before, and fix a nontrivial additive character ψ of F. For α ∈ L, we denote by α¯ its conjugate over F. We fix δ ∈ L × such that δ¯ = −δ and n 0 ∈ F × not contained in N L/F (L × ).


243

For a finite-dimensional L-space W equipped with hermitian or antihermitian form, we denote by U (W ) its unitary group and by GU(W ) its unitary similitude group. For a vector space W with symplectic form, we denote by Sp(W) its symplectic group and by GSp(W) its symplectic similitude group. We denote by Mp(W) the metaplectic group of W. Let V 0 be a finite-dimensional right Fspace with symmetric bilinear form hv, v 0 i F for v, v 0 ∈ V 0 . We denote by SO(V 0 ), O(V 0 ), and GO(V 0 ) the special orthogonal group, the orthogonal group, and the orthogonal similitude group of V 0 respectively. We denote by GO+ (V 0 ) the group of proper similitudes of V 0 . Let V be a finite-dimensional right L-space with hermitian form satisfying hv1 α, v2 βi = αhv ¯ 1 , v2 iβ,

v1 , v2 ∈ V

and let W be a left L-space with antihermitian form satisfying ¯ hαw1 , βw2 i = αhw1 , w2 iβ,

w1 , w2 ∈ W

for α, β ∈ L. Then on W = V ⊗ L W , we can define a symplectic form by hhv1 ⊗ w1 , v2 ⊗ w2 ii = 21 tr L/F hv1 , v2 ihw1 , w2 i . For W, V , we have a dual reductive pair (U (W ), U (V )) in Sp(W). We denote the natural embeddings by ιV : U (W ) → Sp(W), ιW : U (V ) → Sp(W). Assume W is a direct sum of two antihermitian spaces W1 , W2 for L/F, and set

Wi = V ⊗ Wi for i = 1, 2. Similarly as above, we have dual pairs (U (W1 ), U (V )) in Sp(W1 ) and (U (W2 ), U (V )) in Sp(W2 ), and the embeddings

ιV,1 : U (W1 ) → Sp(W1 ), ιW1 : U (V ) → Sp(W1 ), ιV,2 : U (W2 ) → Sp(W2 ), ιW2 : U (V ) → Sp(W2 ). These dual pairs yield the seesaw diagram

(2-1)

U (W )Q

U (V ) × Sp(W2 )

U (W1 ) × {1}

U (V )

QQQ m QQQ mmmmm Q m mmQQ mmm QQQQQ mmm Q

The right vertical line is the map ιW1 × ιW2 : U (V ) → U (V ) × Sp(W2 ) ⊂ Sp(W1 ) × Sp(W2 ).

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HIROSHI SAITO

We recall one more seesaw diagram from [Harris 1993]. Let W 0 be a finitedimensional left F-space with symplectic form h , i F . We can define an antihermitian form on W L0 = L ⊗ F W 0 by X X X 0 αi ⊗ vi , βj ⊗ vj = αi β¯ j hvi , v 0j i F i

j

i, j

for αi , β j ∈ L, and vi . v j ∈ V 0 . Conversely, let V be a right L-space with hermitian form h , i. Then composing the hermitian form with tr L/F , we can define a symmetric bilinear form 0 1 2 tr L/F (hv, v i) on Res F V , the space V considered as an F-space. In this notation we have, from [Harris 1993, (3.5.1.1)],

(2-2)

GU(W L0 )

GO(Res F V )

GSp(W 0 )

GU(V )


3. Application of the theorem of Moen–Rogawski In this section, using the diagram (2-1) and the theorem of Moen–Rogawski, we deduce an analogue of Theorem 1.2 in [Prasad 1994] for unitary groups of degree 2. For α ∈ L × with α¯ = −α, we denote by W (α) the 1 dimensional left L-space L with antihermitian form hx, yi = αx y¯ for x, y ∈ L. For α, β ∈ L × , we set W (α, β) = W (α)⊕ W (β). For a ∈ F × , we denote by V (a) the 1 dimensional right L-space L with hermitian form hx, yi = a x¯ y. We set W = W (δ), W− = W (−δ), and V = V (1), or W = W (n 0 δ), W− = W (−n 0 δ), and V = V (1). Set W = V ⊗ L W , and W− = V ⊗ L W− . Then we have a seesaw diagram of type (2-1): U (W + W− )

U (V ) × Sp(W− )

U (W ) × {1}

U (V )

QQQ QQQ mmm QQmQmmmm m Q mmm QQQQQ mmm Q

We recall the splittings of the above unitary groups into metaplectic groups, following Section 1 of [Harris et al. 1996]. We fix a character χ of L × whose restriction to F × is L/F . Let X be the graph of minus the identity from W to W− , and let Y be the graph of the identity. Then V ⊗ L X and V ⊗ L Y are maximal isotropic subspace of W, and W = V ⊗ L X +V ⊗ L Y yields a complete polarization


245

of W. This determines an isomorphism Mp(W + W− ) ' Sp(W + W− ) o C1 , where the product in Sp(W + W− ) o C1 is given by the Rao cocycle [1993]. The inverse image in Mp(W + W− ) of Sp(W) × {1} or {1} × Sp(W) is isomorphic to Mp(W). By (1.21) of [Harris et al. 1996], we have splittings ι˜V,χ , ι˜V,χ × ι˜V,χ ,− satisfying ι˜V,χ

U (W + W− ) i

- Mp(W + W− ) 6

6

i˜

U (W ) × U (W )


- Mp(W) × Mp(W).

Here we note that U (W− ) = U (W ), Mp(W) = Mp(W− ) and the splitting i˜ : Mp(W) × Mp(W) → Mp(W + W− ) of the embedding i : Sp(W) × Sp(W) → Sp(W + W− ) is specified so that the restriction to central C1 is given by C1 × C1 → C1 ,

(c1 , c2 ) → c1 c¯2 .

Then, by [Harris et al. 1996, Lemma 1.1], (3-1)

ι˜V,χ ,− = χ −1 ι˜V,χ .

In this case, U (V ) is the center of U (W + W− ), and the splitting of U (V ) as the center of U (W + W− ) by χ coincides with the splitting ιW +W− ,χ 2 (Corollary A.8 of the same reference). Let (ωψ , S(V ⊗ L X )) be the Weil representation of Mp(W + W− ) realized on the space of Schwartz–Bruhat functions on V ⊗ F X as the Schrödinger model associated to the complete polarization W = V ⊗ L X + V ⊗ L Y . For a character λ1 of U (V ), let θχ (λ1 , W + W− ) be the theta correspondence of λ1 to U (W + W− ). Namely, let SV,W,χ (λ1 ) be the maximal quotient of S(V ⊗ L X ) on which U (V ) acts as multiple of λ1 . Then SV,W,χ (λ1 ) ' θχ (λ1 , W + W− ) λ1 , as U (W + W− ) × U (V )-spaces with an U (W + W− )-module θχ (λ1 , W + W− ). Let ωψ.W be the Weil representation of Mp(W). Let ψ0 be the additive character of L given by ψ0 (x) = ψ( 21 tr L/F (−δx)) for x ∈ L. For a character η of L 1 , we denote by η L the character of L × given by η L (x) = η(x/x). ¯

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HIROSHI SAITO

Theorem 3.1 (Moen and Rogawski). Let 1 if W = W (δ), = −1 if W = W (n 0 δ). Then ωψ,W ◦ ι˜V,χ |U (W ) =

M

Cη.

ε(χ η−1 L ,ψ0 )=

Remark 3.2. Here we use the character ψ0 instead of ψ ◦ tr L/F . This simplifies some expressions (see Remark in Introduction of [Prasad 1994]). For a character η of U (W ), we denote by θχ (η, V ) the theta correspondence of η in Mp(W) to U (V ). Then θχ (η, V ) = η−1 if η appears in the theta correspondence. We note that U (V ) ' U (W ) ' L 1 , and the embedding ιV and ιW are chosen so that the actions of U (V ) and U (W ) on W are the inverse of each other. By the isomorphism U (V ) ' L 1 , we consider the restriction of χ to L 1 as a character of U (V ) and denote it also by χ. Lemma 3.3. Let the notation be as above. Let U (W ) × {1} be the subgroup of U (W ) × U (W )(⊂ U (W + W− )) consisting of elements with unit in the second component. Then dim HomU (W )×{1} θχ (χ −1 λ1 , W + W− ), η 1 1 if η and λ1 η appear in ωψ,W ◦ ι˜V,χ , = 0 otherwise. Proof. Hom(U (W )×{1})×U (V ) (ωψ , (η 1) χ −1 λ1 ) ' Hom(U (W )×{1})×U (V ) (θχ (χ −1 λ1 , W + W− ) χ −1 λ1 , (η 1) χ −1 λ1 ) ' HomU (W )×{1} (θχ (χ −1 λ1 , W + W− ), η 1). We note that U (V ) is embedded into U (W ) × U (W ) diagonally in Sp(W + W− ), and the action of α ∈ U (V ) for α ∈ L 1 on S(V ⊗ L X ) is given by that of (α −1 , α −1 ) ∈ U (W )×U (W ). By [Mœglin et al. 1987, II.1, Remarques (5), (6)] and [Harris et al. ∨ 1996, Lemma 2.1(i)], the restriction of ωψ to Mp(W) × Mp(W) is ωψ,W ωψ, W. ∨ Here ωψ,W is the contragredient of ωψ,W , and by (3-1) we obtain ∨ ∨ ωψ, W ◦ ι˜V,χ ,− = χ(ωψ,W ◦ ι˜V,χ ) .

Hence Hom(U (W )×{1})×U (V ) (ωψ , (η 1) χ −1 λ1 ) ∨ −1 1 ' HomU (W )×U (V ) η (θχ (η, V ) ⊗ ωψ, λ W ◦ ι˜V,χ ,− ), η χ ∨ −1 1 ' HomU (V ) (θχ (η, V ) ⊗ (χ(ωψ,W ◦ ι˜V,χ , ) ), χ λ .


247

Our assertion follows from this.

Taking λ1 to be the trivial character of L 1 , by Lemma 3.3 and Theorem 3.1, we obtain: Theorem 3.4. Let be as above. Then θχ (χ −1 , W + W− )|U (W )×{1} =

M

Cη 1.

ε(χ η−1 L ,ψ0 )=

Theorem 3.5. Let λ1 be a nontrivial character of L 1 , and let be as above. Then M θχ (χ −1 λ1 , W + W− )|U (W )×{1} = Cη 1. ε(χ (λ1L η L )−1 ,ψ0 ) = ε(χ η−1 L ,ψ0 ) =

4. Prasad’s Theorem We rewrite the results in the previous section in terms of GU(2) and a torus TL in GU(2) isomorphic to L × , and by restricting it to a subgroup of index 2 of GL2 (F), we deduce the theorem of D. Prasad using a seesaw diagram of type (2-2). Let W 0 = F 2 be the two-dimensional left F-space with symplectic form hv1 , v2 i = x1 y2 − y1 x2 for v1 = (x1 , y1 ), v2 = (x2 , y2 ) ∈ W 0 , and let W L0 = L 2 be the two-dimensional left L-space with antihermitian from hv˜1 , v˜2 i H = x1 y¯2 − y1 x¯2 for v˜1 = (x1 , y1 ), v˜2 = (x2 , y2 ) ∈ W L0 . Then we see W (δ, −δ) ' W L0 , as spaces with antihermitian forms. More explicitly, let δ 1/2 h= . −δ 1/2 Then

0 1 h −1 0

t

¯h = δ 0 . 0 −δ

If we take n 0 hv1 , v2 i instead of hv1 , v2 i, we get W (n 0 δ, −n 0 δ) ' W L0 . Similarly, we have 0 n0 t ¯ n δ 0 h h= 0 . −n 0 0 0 −n 0 δ Let Res F V be the two-dimensional right F-space with symmetric bilinear form associated with V (1). For these spaces, we have the following diagram of type

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HIROSHI SAITO

(2-2): GU(W L0 )

GO(Res F V )

GSp(W 0 )

GU(V )


Note that

W0

and V satisfy

GSp(W 0 ) = GL(W 0 ),

Sp(W 0 ) = SL(W 0 ),

SO(Res F V ) = U (V ),

U (W L0 ) ⊃ SU (W L0 ) = SL(W 0 ),

GO+ (Res F V ) = GU(V ).

Let νW (g) be the similitude of g ∈ U (W L0 ). Let GU(W L0 )+ = {g ∈ GU(W L0 ) | L/F (νW (g)) = 1}, GL(W 0 )+ = {g ∈ GL(W 0 ) | L/F (det g) = 1}, and identify L × with the center of GU(W L ). Then GU(W L0 ) ⊃ GU(W L0 )+ = L ×U (W L0 ) = L × GL(W 0 )+ , since N L/F (L × )L ×2 = L 1 L ×2 . Let TL be the torus in GL(W 0 ) isomorphic to L × given by a 2−1 b t 2 t (a, b) ∈ F \ (0, 0) , 2δ 2 b a and let α = a + bδ ∈ L ,

µ = α/α. ¯

We fix the isomorphism a 2−1 b α 7→ 2δ 2 b a

and identify TL with L × . We have µ+1 (2δ)−1 (µ − 1) a 2−1 b α¯ 0 1 . (4-1) = µ+1 2δ 2 b a 0 α¯ 2 (2δ)(µ − 1) We note

1 µ + 1 (2δ)−1 (µ − 1) µ 0 = h −1 h. µ+1 0 1 2 2δ(µ − 1)

We recall the action of some elements on S(V ⊗ L X ). We write them for the pair (U (W L0 ), U (V )). Then X = {(x, 0) | x ∈ L}, Y = {(0, y) | y ∈ L}, and α 0 ∈ U (W L0 ) 0 α¯ −1


249

acts on X by α and on Y by α¯ −1 . Hence α 0 = χ(α¯ −1 ) = χ(α) βV 0 α¯ −1 in the notation of Theorem 3.1 of [Kudla 1994]. By the same theorem we have, for α ∈ L × , α 0 1/2 f (x) = χ(α)|α| L f (αx). ωψ ι˜V,χ 0 α¯ −1 In particular, for α ∈ L 1 , α 0 (4-2) ωψ ι˜V,χ f (x) = χ(α) f (αx). 0 α For the dual pair (SL(W 0 ), SO(Res F V )), let S(λ1 ) be the maximal quotient of S(V ⊗ L X ) = S(Res F V ⊗ F X 0 ), X 0 = {(x, 0) | x ∈ F}, on which SO(Res F V ) acts as multiple of λ1 . Here the action of α ∈ SO(Res F V ) with α ∈ L 1 is given by f (x) 7→ f (α −1 x). Then the above formula implies that SV,W,χ (χ −1 λ1 ) = S(λ1 ). Hence the restriction of the action of U (W L0 ) on the space θχ (χ −1 λ1 , W + W− ) to SL(W 0 ) is the theta correspondence of λ1 to SL(W 0 ). We denote it by θ(λ1 , W 0 ). We extend the theta correspondence θχ of U (V ) to U (W L0 ) to that of GU(V ) to GU(W L0 )+ following [Harris 1993, 3.2]. The similitude νV of GU(V ) satisfies νV (GU(V )) = N L/F L × . Let R(V, W ) = {(g, h) ∈ GU(W L0 ) × GU(V ) | νV (h) = νW (g)}. Then by corresponding (g, h) to the map v ⊗ w 7→ h −1 v ⊗ wg, v ∈ V, w ∈ W L0 , we can takes R(V, W ) into Sp((V ⊗ L W L0 ). We consider a semidirect product U (W L0 ) n GU(V ) defined by hg = hgh with −1 1 0 1 0 . g= g 0 νV (h) 0 νV (h)

h

Then we have an isomorphism R(V, W ) ' U (W L0 ) n GU(V )

250

HIROSHI SAITO

given by −1 1 0 ,h . (g, h) → g 0 νV (h) We let GU(V ) act on S(V ⊗ L X ) by −1/2

L(h) f (x) = χ(α −1 )|α| L

f (α −1 x).

Then L(h) defines a unitary operator on S(V ⊗ L X ), and this action with ωψ ◦ ι˜V,χ defines an action of R(V, W ) on S(V ⊗ L X ) and a splitting of R(V, W ) into Mp(V ⊗ L W L0 ). Let λ be a character of GU(V ) whose restriction to U (V ) is λ1 . We identify λ with a character of L × by GU(V ) ' L × . For a character λ of L × , let λ be the ¯ for α ∈ L × . By the projection to the second character of L × given by λ(α) = λ(α) 0 factor GU(V ) of GU(W L ) × GU(V ), we may see χλ as a character of R(V, W ). Define (S(V ⊗ L X ) ⊗ χ λ)U (V ) to be the maximal quotient of S(V ⊗ L X )⊗χ λ on which U (V ) acts trivially. Then GU(W L0 )+ acts on this space as follows. For g ∈ GU(W L0 )+ , choose h ∈ GU(V ) satisfying νW (g) = νV (h). Define the action of g as that of (g, h) ∈ R(W, V ). Then this is independent of the choice of h. As U (W L0 )-modules, we have (S(V ⊗ L X ) ⊗ χλ)U (V ) ' SV,W,χ (χ −1 λ1 ), and on this space, α0 α0 ∈ GU(W L0 )+ acts by χ λ. We denote the restriction to GL(W 0 )+ of this representation by θ(λ, GL(W 0 )) . Here is as in Section 3. Let a = α α. ¯ Then a 0 α¯ 0 α 0 = 0 1 0 α¯ 0 α¯ −1 Hence α0 01 acts on f˜ ∈ S(λ1 ) sending it to the class in S(λ1 ) of the function 1/2

1/2

χ (α)λ(α)χ ¯ (α)|α| L f (αx) = λ(α)|α| L f (αx). This coincides with the extension of the action of SL(W 0 ) to GL(W 0 )+ in [Jacquet and Langlands 1970, Proposition 1.5]. For a character λ of L × , we set ) 0 + + θ (λ, GL(W 0 )) = Ind GL(W GL(W 0 )+ θ(λ, GL(W ) ) . 0

Then as GL(W 0 )+ -modules, we have ) 0 0 + + 0 + − ResGL(W GL(W 0 )+ θ (λ, GL(W )) = θ(λ, GL(W ) ) ⊕ θ(λ, GL(W ) ) ; 0

see [Mœglin et al. 1987, II.1, Remarque (3)]. Let Ps(1, L/F ) be the principal series representation of GL2 (F) associated with characters (1, L/F ). Then θ (1, GL(W 0 )) is isomorphic to Ps(1, L/F ) by [Jacquet


251

and Langlands 1970, Theorem 4.7]. We set Ps(1, L/F ) the subspace corresponding to θ (λ, GL(W 0 )) . By setting φ = χη−1 L , we see that Theorem 3.4 is equivalent to the following: Theorem 4.1 [Prasad 2007, Lemma 4]. For = ±1, Ps(1, ω L/F ) |TL = ⊕ε(φ,ψ0 )= Cφ, where φ runs through all characters of L × whose restriction to F × is equal to L/F . Remark 4.2. The map η 7→ χ −1 η L induces a one to one correspondence between the set of characters of L 1 and the set of characters of L × whose restriction to F × is L/F . Therefore the theorem of Moen–Rogawski is equivalent to the preceding theorem through Theorem 3.4. For λ such that λ| L 1 is not trivial, θ(λ, GL(W 0 )) is an irreducible supercuspidal representation of GL(W 0 ) by Theorem 4.6 of [Jacquet and Langlands 1970]. In this case, Theorem 3.5 can be stated as follows: Theorem 4.3 [Prasad 1994, Theorem 1.2]. Under the action of GL2 (F)+ , the space θ (λ, GL(W 0 )) decomposes into two subspaces θ(λ, GL(W 0 ))± , and for = ±1, one has M θ (λ, GL(W 0 )) |TL = Cφ, ε(λφ −1 ,ψ0 ) = ε(λφ −1 ,ψ0 ) =

where φ runs through all characters of L × which satisfy λφ −1 | F × = L/F . Proof. Set φ = χ −1 λη L . Since λ1L = λλ−1 , we see −1 χ (η L λ1L )−1 = (χλ−1 η−1 L )λ = λφ , −1 −1 −1 χη−1 L = (χλ η L )λ = λφ .

We note λφ −1 | F × = λφ −1 | F × = L/F . By (4-1), we can see the action of TL by that of L 1 . For v ∈ θχ (χ −1 λ1 , W + W− ), U (W ) × {1} acts on v via η 1 if and only if TL acts on v via χ¯ λη L = χ −1 λη L . The assertion follows from this and Theorem 3.5. 5. Nonsplit case We now consider the nonsplit case. Let α β B= α, β ∈ L . n 0 β¯ α¯

252

HIROSHI SAITO

Then B is the division quaternion algebra over F. Let B + = {x ∈ B | L/F (N (x)) = 1},

B 1 = {x ∈ B | N (x) = 1}.

Here N (x) is the reduced norm of x ∈ B. We set α 0 × TL = α ∈ L . 0 α¯ Then TL ' L × . We note (5-1)

α 0 α¯ 0 α/α¯ 0 = . 0 α¯ 0 α¯ 0 1

Let α = δ, β = −n 0 δ, or α = n 0 δ, β = −n 20 δ. Then B × ⊂ GU(W (α, β)), and TL ⊂ B + ⊂ GU(W (α, β))+ = L ×U (W (α, β)) = L × B + . Here GU(W (α, β))+ is the subgroup of GU(W (α, β)) consisting of elements with similitude in N L/F (L × ). We define splittings. Let W = W (α, −β). We embed W (α, β) into W + W− and consider U (W (α, β)) as a subgroup of U (W + W− ). Let W = V ⊗ F W , and W− = V ⊗ F W− . We may consider W(α, β) = V ⊗ F W (α, β) as a symplectic subspace of W + W− and Sp(W(α, β))) as a subgroup of Sp(W + W− ). Then we have splittings ι˜V,χ , ι˜V,χ,− satisfying ι˜V,χ

U (W + W− ) i

- Mp(W + W− ) 6

6

U (W ) × U (W )

i˜

ι˜V,χ × ι˜V,χ ,−

- Mp(W) × Mp(W).

We choose the embedding of Mp(W)×Mp(W) into Mp(W +W− ) so that it induces the map (c1 , c2 ) 7→ c1 c¯2 on the center C1 × C1 . Let W(α) = V ⊗ L W (α), and W(β) = V ⊗ L W (β). Restricting the above diagram to Mp(W(α, β)), we obtain U (W (α, β)) i

ι˜V,χ

- Mp(W(α, β)) 6

6

U (W (α)) × U (W (β))

i˜


- Mp(W(α)) × Mp(W(β))

Here Mp(W(α, β)) is the inverse image of Sp(W(α, β)) in Mp(W + W− ), and Mp(W(α)) and Mp(W(β)) are the inverse images of Sp(W(α)) and Sp(W(β)) in Mp(W) on the first and the second factor in the above diagram respectively. The


253

restriction of ι˜V,χ : U (W (α, −β)) → Mp(W) to U (W (−β)) induces a map U (W (−β)) = U (W (β))

ι˜V,χ

- Mp(W(−β)) = Mp(W(β)).

Then ι˜V,χ,− and χ −1 ι˜V,χ coincide as homomorphisms of U (W (β)) to Mp(W(β)), by [Harris et al. 1996, Lemma 1.1]. We have ∨ ωψ,W+W ◦ i˜ = ωψ,W(α,−β) ωψ, W(α,−β)

by [Mœglin et al. 1987, II.1 Remarques (5), (6)] and [Harris et al. 1996, Lemma 2.1(i)]. By restricting this to Mp(W(α)) × Mp(W (β)), we obtain ∨ ωψ,W(α,β) ◦ i˜ = ωψ,W(α) χωψ, W(−β) , ∨ ωψ,W(α,β) ◦ i˜ ◦ (˜ιV,χ × ι˜V,χ ,− ) = ωψ,W(α) ◦ ι˜V,χ ωψ, W(−β) ◦ ι˜V,χ ,− ∨ = ωψ,W (α) ◦ ι˜V,χ χωψ,W (−β) ◦ ι˜V,χ

As for the splitting for U (V ), we may take ι˜W +W− ,χ 4 or that induced by ι˜V,χ . Let θχ (χ −1 λ1 , W (α, β)) be the theta correspondence of the character χ −1 λ1 of U (V ) to U (W (α, β)) in Mp(W(α, β)). By the same calculation as in the split case, we obtain: Lemma 5.1. Let U (W (α)) × {1} be the subgroup of U (W (α)) × U (W (β)). Then dim HomU (W (α))×{1} θχ (χ −1 λ1 , W (α, β)), η 1 1 if η appears in ωψ,W(α) ◦ ι˜V,χ and λ1 η appears in ωψ,W(−β) ◦ ι˜V,χ , = 0 otherwise. Since L/F (−β/α) = −1, the trivial character does not satisfy the above condition for λ1 . In the case of a nontrivial λ1 , we have: Theorem 5.2. Let λ1 be a nontrivial character of L 1 , and let = L/F (α/δ). Then M θχ (χ −1 λ1 , W (α, β))|U (W (α))×{1} = Cη 1. −ε(χ (λ1L η L )−1 λ1L ,ψ0 ) = ε(χ η−1 L ,ψ0 ) =

As in the split case, we can interpret this result by the dual reductive pair (B × , GO(V )). In the same way as in the split case, we can define θ(λ1 , B 1 ). Let λ be a character of L × which restriction to L 1 is λ1 . We define the action of L × , the center of U (W (α, β)), on θχ (χ −1 λ1 , W (α, β)) by χλ. Then this yields a well-defined smooth action of L ×U (W (α, β)) on θχ (χ −1 λ1 , W (α, β)), since L × ∩ U (W (α, β)) = L 1 . By restriction, we obtain an action of B + , since B + ⊂ L ×U (W (α, β)). We denote this representation of B + by θ(λ, B + ) for = L/F (α/δ). We induce it to B × and denote it by θ(λ, B × ). By Theorem 5.2 and (5-1), we obtain:

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HIROSHI SAITO

Theorem 5.3. Under the action of B + , θ(λ, B × ) decomposes into two subspaces θ (λ, B × ) for = ±1, and M θ (B × , λ) |TL = Cφ, −ε(λφ −1 ,ψ0 )= ε(λφ −1 ,ψ0 )=

where φ runs through all characters of L × that satisfy λφ −1 | F × = L/F . Remark 5.4. The representations θ(λ, GL(W 0 )) and θ(λ, B × ) are in Jacquet– Langlands correspondence with each other, and Theorem 5.3 gives the latter half of Theorem 1.2 in [Prasad 1994]. By [Mœglin et al. 1987, Chapitre 3, IV, Corollaire 9], an irreducible quotient of θ χ −1 λ1 , W (U (α, β)) is uniquely determined. Since U (W (α, β)) is compact, θ(χ −1 λ1 , U (W (α, β))) is a multiple of this irreducible representation. Lemma 5.1 implies that the multiplicity is 1, and θ (χ −1 λ1 , W (α, β)) is irreducible. Let π = θ(λ, B × ). Since λ| L 1 is not trivial, θ (λ, GL(W 0 )) is supercuspidal. Let π 0 be the representation of B × which corresponds to θ (λ, GL(W 0 )) under the Jacquet–Langlands correspondence. We denote by χπ , χπ 0 the characters of π, π 0 . Then π and π 0 satisfy π ⊗ L/F ' π,

π 0 ⊗ L/F ' π 0 ,

and χπ = χπ 0 on L × . By Corollaries 1.7 and 1.15 of [Hijikata et al. 1993] and Theorem 4.6 (and the remark following it) in [Takahashi 1996], this implies that χπ = χπ 0 on all the other elliptic torus of B × . Therefore π ' π 0 . Acknowledgement The author thanks Professor D. Prasad for calling his attention to these problems. References [Harris 1993] M. Harris, “L-functions of 2 × 2 unitary groups and factorization of periods of Hilbert modular forms”, J. Amer. Math. Soc. 6:3 (1993), 637–719. MR 93m:11043 Zbl 0779.11023 [Harris et al. 1996] M. Harris, S. S. Kudla, and W. J. Sweet, “Theta dichotomy for unitary groups”, J. Amer. Math. Soc. 9:4 (1996), 941–1004. MR 96m:11041 Zbl 0870.11026 [Hijikata et al. 1993] H. Hijikata, H. Saito, and M. Yamauchi, “Representations of quaternion algebras over local fields and trace formulas of Hecke operators”, J. Number Theory 43:2 (1993), 123–167. MR 94e:11126 Zbl 0819.11018 [Jacquet and Langlands 1970] H. Jacquet and R. P. Langlands, Automorphic forms on GL(2), Lecture Notes in Mathematics 114, Springer, Berlin, 1970. MR 53 #5481 Zbl 0236.12010 [Kudla 1994] S. S. Kudla, “Splitting metaplectic covers of dual reductive pairs”, Israel J. Math. 87:1-3 (1994), 361–401. MR 95h:22019 Zbl 0840.22029


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[Mœglin et al. 1987] C. Mœglin, M.-F. Vignéras, and J.-L. Waldspurger, Correspondances de Howe sur un corps p-adique, Lecture Notes in Mathematics 1291, Springer, Berlin, 1987. MR 91f:11040 Zbl 0642.22002 [Moen 1987] C. Moen, “The dual pair (U(3), U(1)) over a p-adic field”, Pacific J. Math. 127:1 (1987), 141–154. MR 88e:22025 Zbl 0675.22008 [Prasad 1994] D. Prasad, “On an extension of a theorem of Tunnell”, Compositio Math. 94:1 (1994), 19–28. MR 95k:22023 Zbl 0824.11035 [Prasad 2007] D. Prasad, “Relating invariant linear form and local epsilon factors via global methods”, Duke Math. J. 138:2 (2007), 233–261. MR 2318284 Zbl 05170260 [Ranga Rao 1993] R. Ranga Rao, “On some explicit formulas in the theory of Weil representation”, Pacific J. Math. 157:2 (1993), 335–371. MR 94a:22037 Zbl 0794.58017 [Rogawski 1992] J. D. Rogawski, “The multiplicity formula for A-packets”, pp. 395–419 in The zeta functions of Picard modular surfaces (Montreal, 1988), edited by R. P. Langlands and D. Ramakrishnan, Publications CRM, Montreal, 1992. MR 93f:11042 Zbl 0823.11027 [Takahashi 1996] T. Takahashi, “Character formula for representations of local quaternion algebras (wildly ramified case)”, J. Math. Kyoto Univ. 36:1 (1996), 151–197. MR 97f:11096 Zbl 0897.22018 Received March 2, 2007. H IROSHI S AITO D EPARTMENT OF M ATHEMATICS FACULTY OF S CIENCE K YOTO U NIVERSITY K YOTO 606-8502 JAPAN [email protected]

ACKNOWLEDGEMENTS The editors gratefully acknowledge the valuable advice of the referees who helped them select and better the papers appearing in volumes 229 through 233 of the Pacific Journal of Mathematics (reports dated June 2006 to December 2007): Ian Agol, Marta Asaeda, Ricardo Baeza, Steven Bell, Marino Belloni, Bruce Berndt, Bernhard Bodmann, Vincent Bonini, Ken Bromberg, Carles Broto, Daniel Bump, Henrique Bursztyn, Mingliang Cai, Danny Calegari, B. Y. Chen, Bin Chen, Sophie Szu-Yu Chen, Xiuxiong Chen, Vladinir Chernov, Raul E Curto, Wenrong Dai, Oliver Dasbach, Josef Diblik, Trond Digernes, Chongying Dong, Jean-Marc Drézet, Olivier Druet, Harold Edwards, Norio Ejiri, Yakov Eliashberg, Matthew Emerton, Charles Epstein, Sam Evens, Jan-Hendrik Evertse, Dashan Fan, Hao Fang, Veronica Felli, Ailana Fraser, Dan Frohardt, Wee Teck Gan, Wee Liang Gan, Yun Gao, Olga Gil-Medrano, Viktor Ginzburg, Sebastian Goette, Rod Gover, Jacob Greenstein, Marco Gualtieri, Fei Han, Zheng-Chao Han, Fengbo Hang, Miloslav Havlicek, Qun He, Zhengxu He, Michael Hoffman, Min-Chun Hong, Timothy Hsu, Zejun Hu, Xiaojun Huang, Mark Huibregtse, Masaki Izumi, William Jaco, Huaiyu Jian, Huiqiang Jiang, Yon Seo Kim, Akitaka Kishimoto, Dexing Kong, Shengli Kong, Motoko Kotani, Marc Lackenby, Kirk Lancaster, Adrian Langer, James Lawrence, Sung-wook Lee, Chris Leininger, Naichung Conan Leung, Jiayu Li, Peter Li, Ren-Cang Li, Tianjun Li, Yanyan Li, Gary Lieberman, C.-G. Liu, Sergei Loktev, Yiming Long, Martin Lorenz, John Lott, Zhiqin Lu, Peng Lu, Xiaonan Ma, Jason Manning, Donald (Tony) Martin, Umberto Massari, John McCuan, Anders Melin, Pengzi Miao, Chris Miller, Claudine Mitschi, Robert Molzon, Frank Morgan, Mohamed Ali Mourou, Dmitri Nikshych, Kaoru Ono, E. M. Opdam, Brad Osgood, Domenico Perrone, Chris Peters, Bob Powers, Xiaofeng Ren, Brooks Roberts, Harold Rosenberg, Wayne Rossman, Yongbin Ruan, Claude Sabbah, Alistair Savage, Marty Scharlemann, Peter Schneider, Richard Evan Schwartz, Freydoon Shahidi, Dimitri Shlyakhtenko, Adam Sikora, Christopher Skinner, M. Solleveld, Ruifang Song, James Sparks, James Spencer, Atanas Stefanov, Robert S. Strichartz, Jiabao Su, Xiaofeng Sun, András Szucs, Iskander Taimanov, Jacques Thévenaz, Gudlaugur Thorbergsson, Dylan Thurston, Mihai Tibar, Tatiana Toro, Valentino Tosatti, Burt Totaro, Khalifa Trimèche, Michela Varagnolo, Monica Vazirani, Misha Verbitsky, Yves Colin de Verdière, Constantin Vernicos, Tom Vogel, Robert Waelder, Changping Wang, Changyou Wang, Jiaping Wang, W. Wang, Xiaodong Wang, Zhiqiang Wang, Lihe Wang, Guofang Wei, N. Wildberger, Siye Wu, Hongwei Xu, Xingwang Xu, G. Yamskulna, Ju-rang Yan, Dmitri Zaitsev, Qi Zhang, Yuxi Zheng, Jiazu Zhou, Xiaohua Zhu, Xiping Zhu.

Volume 234

No. 1

January 2008

Category O for the Virasoro algebra: cohomology and Koszulity B RIAN D. B OE , DANIEL K. NAKANO AND E MILIE W IESNER

1

Convexity in locally conformally flat manifolds with boundary M ARCOS P ETRÚCIO DE A. C AVALCANTE

23

Braid group representations from twisted quantum doubles of finite groups PAVEL E TINGOF , E RIC ROWELL AND S ARAH W ITHERSPOON

33

Interior and boundary regularity of intrinsic biharmonic maps to spheres Y IN B ON K U

43

On two notions of semistability M ARIO M AICAN

69

On a necessary condition for spanners in a wedge S UNG - HO PARK

161

Module supersingulier, formule de Gross–Kudla et points rationnels de courbes modulaires M ARUSIA R EBOLLEDO Two remarks on a theorem of Dipendra Prasad H IROSHI S AITO

167 185

Vol. 234, No. 1

137


2008

Central extensions, symbols and reciprocity laws on GL(n, F) F ERNANDO PABLOS ROMO


PACIFIC JOURNAL OF MATHEMATICS

Volume 234

No. 1

January 2008