L-FUNCTIONS OF GL(2n) : p-ADIC PROPERTIES AND NONVANISHING OF TWISTS
arXiv:1802.10064v1 [math.NT] 27 Feb 2018
MLADEN DIMITROV,
FABIAN JANUSZEWSKI,
A. RAGHURAM
Abstract. The principal aim of this article is to attach and study p-adic L-functions to cohomological cuspidal automorphic representations Π of GL2n over a totally real field F admitting a Shalika model. We use a modular symbol approach, along the global lines of the work of Ash and Ginzburg, but our results are more definitive since we draw heavily upon the methods used in the recent, and separate, works of all the three authors. Our p-adic L-functions are by construction distributions on the Galois group of the maximal abelian extension of F unramified outside p∞. Moreover, we prove the so-called Manin relations between the p-adic L-functions at all critical points. This has the striking consequence that, given a Π admitting at least two critical points, and given a prime p such that Πp is ordinary, the central critical value L( 12 , Π ⊗ χ) is not zero for all except finitely many Dirichlet characters χ of finite order which are unramified outside p∞.
Contents 1. Introduction 2. Local considerations 2.1. Some notation 2.2. Towards the distribution relation – local aspects 2.3. Twisted local Shalika integrals 3. Automorphic cohomology 3.1. Group-theoretic notation 3.2. Lattices in finite-dimensional representations 3.3. Automorphic Cohomology and Hecke action 4. Distributions attached to cohomology classes on GL(2n) 4.1. Automorphic symbols: the maps τβ and ιβ 4.2. The maps κj 4.3. A degree-zero class 4.4. Summing over the second component 4.5. The distribution µφj,η 4.6. Manin relations 5. L-functions for GL(2n) 5.1. Representations with a Shalika model 5.2. Local aspects: p-ordinarity and nonvanishing of a local zeta integral 5.3. p-adic interpolation of critical values 5.4. Nonvanishing of twists References
2 4 4 4 5 10 10 10 12 17 17 18 19 20 20 22 26 26 30 34 37 39
Date: February 28, 2018. 2010 Mathematics Subject Classification. Primary: 11F67; Secondary: 11F41, 11F66, 11F70, 11F75, 22E50, 22E55. 1
2
MLADEN DIMITROV,
FABIAN JANUSZEWSKI,
A. RAGHURAM
1. Introduction A crucial result in Shimura’s work on the special values of L-functions of modular forms concerns the existence of a twisting character to ensure that a twisted L-value is nonzero at the center of symmetry; see [Sh, Thm. 2]. Since then it has been a very important problem in the analytic theory of automorphic L-functions to find twisting characters to render a twisted L-value nonzero. Rohrlich [Ro] proved such a nonvanishing result in the context of cuspidal automorphic representations of GL(2) over any number field. This was then generalized to GL(n) over any number field by Barthel-Ramakrishnan [BR] and further refined by Luo [L]; however, neither [BR] nor [L] can prove this at the center of symmetry if n ≥ 4. (For us the functional equation will be normalized so that s = 1/2 is the center of symmetry.) There have been other analytic machinery that has been brought to bear on this problem, see for example, Chinta-Friedberg-Hoffstein [CFH]. Even for simple situations involving L-functions of higher degree this problem is open; for example, suppose ϕ is a holomorphic cusp form, and π = π(ϕ) is the associated cuspidal automorphic representation, then it has been an open problem to find a character χ such that for the twisted symmetric cube L-function one has L( 21 , Sym3 π ⊗ χ) 6= 0. In this article, for a cohomological unitary cuspidal automorphic representation Π on GL(2n) over a totally real field F , under a very mild regularity assumption on the infinity type that ensures at least two critical points one of which is the center of symmetry for the standard L-function of Π, supposing Π admits a Shalika model, then for any ordinary prime p for Π, we prove L( 12 , Π ⊗ χ) 6= 0 for all but finitely many Dirichlet characters χ of p-power conductor. Furthermore, we can prove simultaneous nonvanishing results: suppose for 1 ≤ j ≤ r, Πj is a representation of GL(2nj ) as above, and p is ordinary for all of them, then L( 21 , Π1 ⊗ χ) · · · L( 12 , Πr ⊗ χ) 6= 0 for all but finitely many Dirichlet characters χ of p-power conductor. (For example, with a classical normalization of L-functions, it follows from our results that there are infinitely many Dirichlet characters χ such that L(6, ∆⊗χ)L(17, Sym3 ∆⊗χ) 6= 0 for the Ramanujan ∆-function.) Our methods are purely arithmetic and involve studying p-adic distributions on the Galois group of the maximal abelian extension of F unramified outside p and ∞ that are attached to suitable eigen-classes in the cohomology of GL(2n). Let’s now describe our methods and results in more detail. We begin with a purely cohomological situation, without any reference to automorphic forms or L-functions. Let O be the ring of integers of a totally real field F . Take a pure dominant integral weight µ for G = resO/Z GL(2n) (for terms that are not defined in the introduction, the reader will have to consult the main body of the paper), and let Vµ,E be the algebraic irreducible representation of G × E, for some ‘large enough’ p-adic field E; large enough means that E contains a copy of every Galois conjugate of F and also the values of some characters which for the purposes of the introduction we may ignore. For any open compact subgroup K ⊂ G(A(∞) ) of the group of finite adeles, let Vµ,E be the sheaf on the locally symmetric G of G with level structure K given by V space SK µ,E and let’s consider the cohomology • G H (SK , Vµ,E ). If O is the ring of integers of E, and Vµ,O is an O-lattice stabilized by G(O) G, V then we also consider integral cohomology H • (SK µ,O ). As usual there is a Hecke action on these cohomology groups, and we consider an eigenclass φ for a particular Hecke operator G, V Up◦ acting on Hcq0 (SK µ,O ); where the cohomology-degree q0 is the dimension of a locally
p-ADIC L-FUNCTIONS AND NONVANISHING
3
symmetric space for H = resO/Z (GLn × GLn ) ⊂ G. The weight µ determines a contiguous string of integers Crit(µ) (which would correspond to the the set of critical points for an L-function) and for each j ∈ Crit(µ) we attach an O-valued distribution µφ,j on CF (p∞ ), the Galois group of the maximal abelian extension Fp,∞ of F which is unramified outside p and ∞. In Sect. 4 we construct and study these distributions, and especially prove the Manin relations which give a relation between µφ,j and µφ,j ′ for j, j ′ ∈ Crit(µ). See the diagram in (40) to get a quick overview of the sheaf-theoretic maps that are needed to define the distribution µφj,η on CF (p∞ ) attached to φ and j as above. Manin relations say that if j, j + 1 ∈ Crit(µ) then we have an equality of distributions µφ,j+1 = χcyc F · µφ,j on CF (p∞ ); here χcyc F is the p-adic cyclotomic character of F . See Thm. 4.2. Now we apply the above considerations to the situation when φ is related to a cuspidal automorphic representation Π of G(A) which has cohomology with respect to µ; we denote this as Π ∈ Coh(G, µ); see Sect. 5.1.4. Friedberg and Jacquet related the period integral of cusp forms in Π over the subgroup H to the standard L-function L(s, Π), and for the unfolding of this integral to see Eulerian property, the representation is assumed to have a Shalika model; this is reviewed in Sect. 5.1. This anaytic theory for L(s, Π) may be seen as map coming from a Poincar´e duality pairing, and such a cohomological interpretation allows one to deduce algebraicity results for the critical values of the twisted L-functions L(s, Π ⊗ χ). If we take φ = φΠ a cohomology class related to Π, then the considerations of Sect. 4 give us p-adic interpolation properties, i.e., the distribution µΠ,j on C(p∞ ), associated to φΠ and to a critical point 21 + j for L(s, Π), when evaluated on a character χ of conductor pβ gives us essentially the algebraic part of the critical value L( 21 , Π ⊗ χ). See Thm. 5.13. Manin relations when applied to µΠ,j gives congruence relations between two successive critical values. Now suppose these successive L-values were at the center 3 1 2 and the next one which would be 2 ; then since the complex L-function of a unitary cuspidal representation doesn’t vanish for ℜ(s) ≥ 1 we deduce from the congruence that the distribution corresponding to the center of symmetry is also nonvanishing, which in turn gives us the nonvanishing results mentioned above. See Thm. 5.15, Thm. 5.17, and Cor. 5.18. Let’s mention some relevant papers in the literature. First of all, Ash and Ginzburg [AG] started the study of p-adic L-functions for GL(2n) over a totally real field by considering the analytic theory studied by Friedberg and Jacquet [FJac]. However to quote the authors of [AG], their results are definitive only for GL(4) over Q and for cohomology with constant coefficients. Furthermore, they construct their distributions on local units while only suggesting that one should really work, as we do in this paper, on the Galois group of Fp,∞ /F (see also the issues raised in [Jan1]). This article uses the more recent results and techniques developed in independent papers by all the three authors; namely, [Di], [GRa2], and [Jan2, Jan3]. Finally, we mention Gehrmann’s thesis [Ge] which also constructs p-adic L-functions in essentially a similar context; however, his methods are entirely different from ours. To conclude the introduction, our emphasis is on the purely sheaf-theoretic nature of the construction of the distributions attached to eigenclasses in cohomology and this leads to a purely algebraic proof of Manin relations in a very general context. When specialized to a cohomology class related to a representation Π of GL2n , we get p-adic interpolation of the
4
MLADEN DIMITROV,
FABIAN JANUSZEWSKI,
A. RAGHURAM
critical values of the standard L-function L(s, Π), and Manin relations give nonvanishing of twists L(s, Π ⊗ χ) at the center of symmetry. A nonvanishing theorem in the realms of analytic number theory admitting a decidedly algebraic proof is philosophically piquant. Acknolwedgements: This project started when the three of us met at a conference in July 2014 at IISER Pune on p-adic aspects of modular forms. Any subset of two of the authors is grateful to the host institute or university of the third author during various stages of this work. MD and AR are grateful to an Indo-French research grant from CEFIPRA that has facilitated visits by each to the work-place of the other.
2. Local considerations In this section we delineate some local calculations that will be needed in the main global calculations. For a first reading, the reader should skip this section and start with Sect. 3, and come back to this section as and when necessary. In this section, F/Qp denotes a non-archimedean local field of residue characteristic p > 0, O ⊆ F the valuation ring with maximal ideal p and ̟ a fixed uniformizer. 2.1. Some notation. For n ≥ 2, let Tn denote the diagonal subgroup of GLn . Put H := GLn × GLn ⊆ GL2n , i.e., elements of H are block diagonal matrices diag(h1 , h2 ) with h1 , h2 ∈ GLn . We write deti (diag(h1 , h2 )) = det hi , i ∈ {1, 2}. Let Mn (F) denote the matrix algebra of n × n-matrices over F. For m ∈ Mn (F ), let u(m) = 1n 1mn ∈ GL2n (F ). We let In (resp. In− ) denote the standard upper (resp. lower) triangular Iwahori subgroup in GLn (O). The following two matrices play an important role in local calculations: ̟1n 1n wn t̟ := , and ξ := , 1n wn where 1n and 0n are the n × n identity and zero matrices, respectively; and wn is the Weyl group element of longest length in GLn , whose (i, j)-entry is wn (i, j) = δi,n−j+1 . (We will often drop thesubscript n from 1n and 0n and simply write 1 and 0 instead.) We have n ξ −1 = 1n −1 wn . Once and for all we record the identities A B A − C (A − D + B − C)wn −1 ξ · ·ξ = C D wn C (C + D)wn and ξ·
A B A + wn C D wn − A + Bwn − wn C · ξ −1 = . C D wn C D wn − wn C
2.2. Towards the distribution relation – local aspects. We lay ground for the proof of the distribution relation in Section 4. We prove a couple of local statements that would interrupt the flow of the proof of Theorem 4.1. Lemma 2.1. For each β ≥ 1 and m ∈ Mn (O), put km := 1 + ̟ β mwn . We have: −1 km km β · ξt̟ u(m)t̟ · = ξtβ+1 ̟ . 1n 1n Furthermore, the map m 7→ det(km ) (mod (1 + p(β+1) )) from Mn (O) to (1 + pβ )/(1 + pβ+1 ) is a surjective group homomorphism.
p-ADIC L-FUNCTIONS AND NONVANISHING
5
Proof. To verify the matrix identity, rewrite it as km −1 −(β+1) −1 km t̟ u(m)t̟ = t̟ ξ ξtβ+1 ̟ , 1n 1n and now use the identity in Sect. 2.1 for conjugating by ξ −1 , and then simplify. The last assertion follows from det(km ) ≡ 1 + ̟ β tr(mwn ) (mod (1 + pβ+1 )). Set for β ≥ 0, −1 Lβp := H(F ) ∩ ξtβ̟ I2n t−β ̟ ξ .
We’ll need the following lemma on the structure of Lβp . Lemma 2.2. For any β > 0, we have k β wn : k ∈ In ∩ In , m ∈ Mn (O) . Lp = k − ̟β m × × Under the homomorphism det1 det−1 2 × det2 : H(F ) → F × F , that sends diag(h1 , h2 ) to (det(h1 h−1 2 ), det(h2 )) we have β (1) (det1 det−1 1 + pβ × O× . 2 × det2 )(Lp ) =
Furthermore,
Lβ+1 ⊆ Lβp p
(2) and
(Lβp : Lβ+1 ) = #Mn (O/p). p
(3)
β −1 k1 Proof. Let k1 k2 ∈ Lβp . By definition, we have t−β ̟ ξ k2 ξt̟ ∈ I2n . This means k1 ̟ −β (k1 −k2 )wn ∈ I2n , which is the case if and only if k1 , k2wn ∈ In and k1 − k2 ∈ that k wn 2
̟ β Mn (O). Take k1 = k ∈ In and for m ∈ Mn (O) take k2 = k − ̟ β m, then k2wn ∈ In is guaranteed when k ∈ In ∩ Inwn . (We could have also taken k2 = k − ̟β mwn .) From this description of Lβp , the remaining assertions are obvious.
2.3. Twisted local Shalika integrals. We will review the global theory of Shalika models and L-functions in detail later in the article, but in this section, we evaluate a particular local zeta integral that will be used in the global interpolation formulas. 2.3.1. Local Shalika vectors. Fix an additive character ψ : F → C× of conductor O and a multiplicative character η : F × → C× . Let W denote a (right) Iwahori invariant local (η, ψ)-Shalika vector on GL2n (F ), i.e., W is a C-valued function on GL2n (F ) such that h 1n X (4) W g k = η(det h)ψ(tr X)W (g), h 1n for h ∈ GLn (F ), X ∈ Mn (F ), g ∈ GL2n (F ) and k ∈ I2n .
6
MLADEN DIMITROV,
FABIAN JANUSZEWSKI,
A. RAGHURAM
2.3.2. Local Birch Lemma. Let α ≥ 1 and denote by χ : F × → C× a quasi-character of conductor fχ = ̟ α O. Consider the twisted local zeta integral Z 1 g α W ζ(s; W, χ) := ξt̟ χ(det g)| det g|s− 2 dg 1 GLn (F ) for s ∈ C with ℜ(s) ≫ 0. Our aim is to establish Theorem 2.3 (Local Birch Lemma). Let W be a Iwahori invariant Shalika vector on GL2n (F ). Then there is a non-zero rational constant c, only depending on F and n, with the property that for every finite order character χ : F × → C× with non-trivial conductor fχ , 1
ζ(s; W, χ) = c · G(χ)n · N(fχ )n(s− 2 )−
n(n+1) 2
· W (diag(1n , wn ))
for all s ∈ C with ℜ(s) ≫ 0. Let’s remark that in applications, we would like the local zeta integral in the above theorem to be nonzero. This boils down to W (diag(1n , wn )) 6= 0 where W is an Iwahorispherical Shalika function on GL2n (F ). We will take this up in Sect. 5.2.3. The proof of Theorem 2.3 will occupy several subsections. 2.3.3. First reductions. For any g ∈ GLn (F ), observe that g g α α α (5) W ξt̟ = ψ(̟ tr gXwn ) · W ξt̟ , 1 1
X ∈ Mn (O),
The first relation follows by right Iwahori invariance and the Shalika property (4): g g 1 X α α W ξt̟ = W ξt̟ = 1 1 1 g 1 ̟α Xwn g α α α W ξt̟ = ψ(tr ̟ gXwn ) · W ξt̟ . 1 1 1 Next, for g ∈ GLn (F ), observe that g g α (6) W ξt̟ = ψ(tr g) · W tα . 1 wn ̟ This follows directly from the Shalika property (4): g g gwn α 1n g g α W ξt̟ = W t̟ = W tα 1 wn 1n wn ̟ g = ψ(tr g) · W Proposition 2.4. We have Z ζ(s; W, χ) =
GLn (F )∩̟ −α Mn (O)
W
g wn
tα̟
wn
tα̟
.
1
· χ(det g)ψ(tr g) · | det g|s− 2 dg
Proof. If g ∈ GLn (F ) has an entry gi0 j0 satisfying |gi0 j0 | > |̟ −α |, then we find an elementary matrix E ∈ Mn (O) with (gE)ij = gi0 j if i = j0 and (gE)ij = 0 if i 6= j0 . Therefore,
p-ADIC L-FUNCTIONS AND NONVANISHING
7
multiplying E by an appropriate unit in O× produces a matrix Y ∈ Mn (O) satisfying ψ(̟ α tr gY ) 6= 1. Hence, putting X := Y wn in (5) we get W (( g 1 ) ξtα̟ ) = 0, whence Z 1 g α W ζ(s; W, χ) = ξt̟ χ(det g)| det g|s− 2 dg. 1 −α GLn (F )∩̟ Mn (O) Application of (6) then proves the proposition.
2.3.4. Torus actions and Gauß sums. Consider the decomposition G GLn (F ) = Bn (F ) · GLn (O) = Un (F ) · ̟ e ωIn , ω∈Sn ,e∈Zn
from which we get
G
GLn (F ) ∩ Mn (O) =
(Un (F )̟ e ∩ Mn (O)) · ωIn .
ω∈Sn ,e∈Zn ≥0
Now Un (F )̟ e ∩ Mn (O) consists of the matrices u = (uij ) ∈ Bn (F ) satisfying i < j, O, (7) uij ∈ {̟ ei }, i = j, {0}, i > j.
A subset of these elements u yields, together with w ∈ Sn , a system of representatives Un ⊆ Gn (F ) of the right In -orbits in GLn (F ) ∩ Mn (O). Proposition 2.5. For e ∈ Zn≥0 , ω ∈ Sn and u ∈ Un (F ) satisfying (7), the non-vanishing of the integral Z g W (8) tα̟ χ(det g)ψ(tr g)dg w −α n ̟ uωIn
implies e = 0 and ω = 1. In the latter case we have uωIn = In and Z 1 g W tα̟ χ(det g)ψ(tr g)| det g|s− 2 dg = wn ̟ −α In Z n Y × −n n nα( 12 −s) (O : 1 + fχ ) G(χ) |̟| χ(rii )−1 dr. χ(det r) W (diag(1n , wn )) · In
i=1
Proof. By Iwahori invariance, non-vanishing of (8) implies non-vanishing of Z χ(det g)ψ(tr g)dg. (9) ̟ −α uωIn
We may rewrite this integral as Z Z χ(det ̟ −α uωg)ψ(tr ̟ −α uωg)dg χ(det g)ψ(tr g)dg = In ̟ −α uωIn Z Z χ(det ̟ −α uωtr)ψ(tr ̟ −α uωtr)dtdr. = In
Tn (O)
We claim that the partial integral Z χ(det ̟ −α uωtr)ψ(tr ̟ −α uωtr)dt (10) Tn (O)
8
MLADEN DIMITROV,
FABIAN JANUSZEWSKI,
A. RAGHURAM
vanishes unless ω = 1 and e = 0. This will imply the proposition. We proceed by induction on n, the case n = 0 being clear. Suppose n > 0 and the claim is true for n − 1. Writing ω t = diag(t ω(1) , . . . , tω(n) ), we have u11 tω(1) u12 tω(2) . . . . . . u1n tω(n) . 0 u22 tω(2) . . u2n tω(n) .. .. .. .. .. uωt = ·ω = . . . . . .. .. .. .. . . . . 0 ... . . . 0 unn tω(n) u1ω−1 (1) t1 . . . u11 tσ(1) . . . u1ω−1 (n) tn u2ω−1 (1) t1 . . . 0 . . . u2ω−1 (n) tn , .. .. .. . ... . ... . unω−1 (1) t1 . . .
0
. . . unω−1 (n) tn
where in the last matrix u11 tσ(1) is located in the σ(1)-th column. Assume σ(1) > 1. Right multiplication by r = (rij ) ∈ In yields a matrix uωtr with entries (uωtr)ij independent of u11 , tσ(1) and r1σ(1) whenever i > 1, and (uωtr)1j = u1ω−1 (1) t1 r1j + · · · + u11 tσ(1) rσ(1)j + · · · + u1ω−1 (n) tn rnj . Therefore, on the one hand, ψ(tr ̟ −α uωtr) = ψ(̟ −α u11 tσ(1) rσ(1)1 ) · ψ(u] ωtr) · ψ(˜ x), where, u] ωtr ∈ Mn−1 (F ) denotes the lower right (n − 1) × (n − 1) block of uωtr and n X
x ˜ :=
u1ω−1 (k) tk rk1
k=1,k6=σ(1)
independent of u] ωtr, tσ(1) and r1σ(1) . On the other hand, χ(det ̟ −α uωtr) = χ(̟ −α u11 tσ(1) ) · χ(det u] ωtr). Now σ(1) > 1 implies rσ(1)1 ∈ p, whence Z χ(det ̟ −α uωtr)ψ(tr ̟ −α uωtr)dtσ(1) = × O Z ] ] χ(̟ −α u11 tσ(1) )ψ(̟ −α u11 tσ(1) rσ(1)1 )dtσ(1) = 0. χ(det uωtr)ψ(uωtr)ψ(˜ x) · O×
Therefore, (10) vanishes, unless σ(1) = 1. Likewise, the same computation shows that non-vanishing of (10) implies e1 = 0. In this case we may unfold Z χ(det ̟ −α uωtr)ψ(tr ̟ −α uωtr)dt = Tn (O) Z Z χ(det ̟ −α u ˜ω ˜ t˜)ψ(tr ̟ −α u ˜ω ˜ t˜r˜)dt˜, χ(̟ −α u11 t1 )ψ(tr ̟ −α u11 r11 t1 )dt1 χ(det r)· O×
Tn−1 (O)
with g˜ ∈ Mn−1 (F ) denoting the lower right n − 1 × n − 1 block of a matrix g ∈ GLn (F ). The induction hypothesis then implies that the latter integral can be non-zero only for ω ˜ = 1n−1 and e = 0, whence the first claim follows.
p-ADIC L-FUNCTIONS AND NONVANISHING
9
In order to evaluate (8), we evaluate (10). For any r ∈ I, Z χ(det ̟ −α tr)ψ(tr ̟ −α tr)dt = Tn (O)
χ(det r) ·
n Y i=1
−1 χ r11 ·
Z
n Y
χ(̟ −α ti rii )ψ(̟ −α ti rii )dt =
Tn (O) i=1
χ(det r) ·
n Y i=1
This concludes the proof of Prop. 2.5.
−1 χ r11 · (O× : 1 + fχ )−n · G(χ)n .
2.3.5. Partial integrals. Let’s introduce the subset In∆=1 = {r ∈ In | rii = 1, ∀i } . Then
Z
χ(det r)
In
n Y
−1
χ(rii )
dr =
Z
χ(det x)dx,
∆=1 In
i=1
where dx = ⊗i6=j dxij is a suitably normalized measure built out of additive Haar measures. Lemma 2.6. The integral
Z
O n(n−1)/2
χ(x) ⊗i>j dxij
vanishes unless xij ∈ fχ for all 1 ≤ j < i ≤ n. Proof. Consider an n × n-matrix x = (xij ) ∈ Mn (O), with upper left (n − 1) × (n − 1)-block x′ and xii = 1 for 1 ≤ i ≤ n : 1 x1,2 ... x1,n−1 x1,n .. .. .. .. ′ . . . x2,1 ′ x . x = , x = . . . . .. .. .. xn−1,n xn−2,n−1 xn,1 . . . xn,n−1 1 xn−1,1 . . . xn−1,n−2 1
We already know that we may assume xij ∈ p for i < j. Assume that for some i0 > j0 we have xi0 j0 6∈ fχ and that i0 is maximal with this property. Then the value of χ(det x) is independent of the entries xij characterized by i > i0 or j > i0 . Therefore, we may assume without loss of generality that i0 = n and appeal to the previous decomposition of x into a square matrix x′ and the entries in the last column and last row. For all integrands xij , ∆=1 ⊆ GL i < j, the matrix x′ lies in In−1 n−1 (O). Therefore, we find elements α1 , . . . , αn−1 ∈ p satisfying n−1 X αi xij = −xnj , i=1
for all 1 ≤ j ≤ n − 1. Adding for 1 ≤ i ≤ n − 1 the αi -multiple of i-th row of x onto the last row, we obtain a n × n-matrix y, whose first n − 1 rows agree with that of x and whose last row has zero entries except at n, where we find ynn = 1 +
n−1 X i=1
αi xin ∈ 1 + p.
10
MLADEN DIMITROV,
FABIAN JANUSZEWSKI,
A. RAGHURAM
Then det x = ynn det(x′ ). By our assumption, |fχ | < |αi0 | < 1, whence when xi0 n runs through O, yn,n runs through a translate of a non-trivial subgroup of O× /(1 + fχ ). Therefore, by the non-triviality of χ on the latter group, Z O
χ(det x)dxi0 n = 0.
Arranging the order of integration accordingly, shows that the integral in the statement of the lemma vanishes. Proposition 2.7. There is a rational constant c ∈ Q× , only depending on n and F, such that Z n(n−1) χ(det r)dr = c · N(fχ )− 2 . ∆=1 In
Proof. For any r ∈ In satisfying rij ∈ fχ , for all i < j, we have det r ∈ 1 + fχ . By the previous discussion we know that there is some rational constant c ∈ Q× satisfying Z Z Z Y Y dxij dxij χ(det g)dg = c · n(n−1) χ(det x) n(n−1) ∆≡1 In
= c· = c·
Z
Z
O
2
n(n−1) 2
fχ
n(n−1) fχ 2
Z Z
O
O
2
n(n−1) 2
n(n−1) O 2
i>j
i 1, there is no canonical identifiG [c]; V K [c]) with an element in V. In order to overcome this cation of an element in H 0 (SK ambuigity, we introduce for γ ∈ G(Q) the map ¯ c ; V) → H∗q (γ Γ ¯ c γ −1 ; V), s 7→ γ · s. tΓc ,γ : H∗q (Γ We set for g ∈ G(A(∞) ) Γgc := G(Q) ∩ d(c)gKg −1 d(c)−1 , the intersection taking place in G(A(∞) ). With this notation we have (∞)×
Proposition 3.2. For any c, c′ ∈ AF
and any g ∈ G(A(∞) ) with
[c′ ] = [det(g)c] ∈ C(det(K)) there exists γ ∈ G(Q) such that γf ∈ d(c)gKd(c′ )−1 . Any such element renders the following diagram commutative: ¯ c′ ; Vc′ ) H∗q (Γ ∼ = y(23)
tΓ
,γ
−−−c−→
tK,g
¯ gc ; Vc ) H∗q (Γ ∼ (23)y=
.
K [c]) G [c′ ], V K [c′ ]) − G H∗q (SK −−−→ H∗q (SgKg −1 [c]; V
The discussion above concerning locally symmetric spaces, local systems, sheaves and cohomology of arithmetic groups are all also applicable for H instead of G.
p-ADIC L-FUNCTIONS AND NONVANISHING
17
4. Distributions attached to cohomology classes on GL(2n) For the benefit of the reader,Q let’s recall our standing assumptions on the choice of level structure. Let Fp = F ⊗Q Qp = p|p Fp . For each p, recall that Ip (resp., Jp ) is the standard Q Iwahori (resp., parahoric of type (n, n)) subgroup of GL2n (Fp ). Let Ip := p|p Ip and Q (∞) Jp := ) be an open compact subgroup such that: p|p Jp . We let K ⊂ G(A
G is a smooth manifold; (1) K is sufficiently small, so that SK (2) at the prime p, Ip ⊆ Kp ⊆ Jp for all p | p; hence, Ip ⊆ Kp ⊆ Jp ; (3) later, this p will be outside a finite set SΠ of finite primes where a cuspidal representation Π is ramified.
4.1. Automorphic symbols: the maps τβ and ιβ . 4.1.1. Locally symmetric spaces for H. For any open-compact subgroup L ⊂ H(A(∞) ) let’s denote the locally symmetric space: (24) S˜H := H(Q)\H(A)/(H∞ ∩ K ◦ Z ◦ )◦ L. ∞
L
∞
To understand the group at infinity that we are quotienting out on the right, at each infinite place of F , we have: ◦ GLn (R) 0 SO(n) 0 ◦ = ∩ (SO(2n)Z2n (R) ) Z2n (R)◦ . 0 GLn (R) 0 SO(n)
It’s easy then to see that (25)
dim(S˜LH ) = [F : Q](n2 + n − 1) =: q0 .
G in degree q , and pull it back to S ˜H for a suitable L, If we consider a cohomology class on SK 0 L then we end up with a top-degree class. The degree q0 happens to be top-most degree with G . This magical numerology is at the heart of what nonvanishing cuspidal cohomology of SK ultimately permits us to give a cohomological interpretation to an integral representing an L-value (see [GRa2]), and so permitting us to study it’s p-adic properties.
4.1.2. The map ιβ . Let β = (βp )p|p be a multi-exponent given by non-negative integers Q β βp , p | p; we will be looking at congruences, distributions, etc., modulo pβ := p|p ̟p p for fixed uniformizers ̟p ∈ p ⊆ Op as before. Recall the elements tβp ∈ G(Qp ) from (12) and the semi-group ∆+ p they generate (cf. (13)). Consider the element β 1 wn p 1 β ˆ ⊂ G(A) ξtp = ∈ G(Z) 0 wn 1
as an element in G(A) by extension via identity elements at all places other than p, and β at p it is ξtβp in G(Qp ), i.e., it is ξt̟pp in GL2n (Fv ) for p | p. Consider an open compact subgroup Lβ ⊂ H(A(∞) ) Q with the following property: We have a factorization Lβ = q Lβ,q into open compact subgroups Lβ,q ⊆ H(Fq ) and at all places q 6= p we take Lβ,q ⊂ Kq independent of β, and at q = p | p we take it as β
−β
Lβ,p := Lβp = H(Fp ) ∩ ξt̟pp Ip t̟p p ξ −1 .
18
MLADEN DIMITROV,
FABIAN JANUSZEWSKI,
A. RAGHURAM
Then the map (26)
G , ιβ : S˜LHβ → SK
[h] 7→ [hξtβp ],
−1 L ξtβ ⊂ K. We assume that outside of p, L is chosen is well-defined, i.e., we have t−β p ξ β p small enough such that its associated arithmetic subgroups are torsion free mod center, i.e. S˜LHβ is a manifold for all β satisfying βp ≥ 1, p | p. Furthermore, using a well-known result of Borel and Prasad (see, for example, [A, Lem. 2.7]) we get that ιβ is a proper map. G to S ˜H : Hence we can pull-back cohomology with compact supports from SK Lβ
(27)
G ; Vµ,O ) −→ Hc• (S˜LHβ ; ι∗β Vµ,O ), ι∗β : Hc• (SK
which we will consider especially in degree • = q0 as in (25). 4.1.3. The map τβ . By our choice of Lβ as in Sect. 4.1.2 above, we see that the inclusion ι : H ֒→ G induces a map, also denoted ι: G . ι : S˜LH → SK β
(Note the subtle difference between ι and ιβ .) The map ι is also proper. Now, we can G via ι. Towards this, we have by the pull back (compactly-supported) cohomology of SK K equivariance of the sheaf Vµ,E a canonical isomorphism of sheaves: (28)
τβ : ι∗β Vµ,O −→ ι∗ ρµ (ξtβp )Vµ,O ,
which together with the • action induces an isomorphism, µ∨ (tβp )τβ : ι∗β Vµ,O −→ ι∗ ρµ (ξ)tβp • Vµ,O , which with the embedding i as in (20) induces a monomorphism (29)
τβ,O : ι∗β Vµ,O −→ ι∗ Vµ,O ,
which induces a map in cohomology ∗ : Hc• (S˜LHβ , ι∗β Vµ,O ) −→ Hc• (S˜LHβ , ι∗ Vµ,O ). (30) τβ,O Putting (27) and (30) together, we get a map: (31)
∗ G ι∗β : Hc• (SK τβ,O ; Vµ,O ) −→ Hc• (S˜LHβ , ι∗ Vµ,O ),
where as mentioned before, we will consider this especially in degree • = q0 . 4.2. The maps κj . Given a dominant integral pure weight µ ∈ X0∗ (T ), define the set (32)
Crit(µ) := {j ∈ Z : µτ,n ≥ j ≥ µτ,n+1 , ∀τ : F → Ω}.
The motivation for this definition comes from the fact proved in [GRa2, Prop. 6.1] that if Π is a cuspidal automorphic representation of GL2n /F , and Π has cohomology with respect to µ–see Sect. 5.1.4 below–then 12 + j with j ∈ Z is critical for standard L-function L(s, Π ⊗ χ) for any finite order character χ if and only if j ∈ Crit(µ). For r1 , r2 ∈ Z define V(r1 ,r2 ) as the one-dimensional representation of H: diag(h1 , h2 ) 7→ NF/Q (det(h1 )r1 det(h2 )r2 ). Next, it follows from [GRa2, Prop. 6.3] that for j ∈ Crit(µ) we have (33) dim HomH (Vµ , V(j,w−j) ) = 1.
p-ADIC L-FUNCTIONS AND NONVANISHING
19
Fix a nonzero κj ∈ HomH (Vµ , V(j,w−j) ) normalized so as to get an integral map: κj : Vµ,O → V(j,w−j),O . For each j ∈ Crit(µ), we get a map in integral-cohomology (34)
κ∗j : Hc• (S˜LHβ , ι∗ Vµ,O ) −→ Hc• (S˜LHβ , V(j,w−j),O ).
4.2.1. A summary of assumptions on µ. For the benefit of the reader, let’s summarize the standing assumptions on a weight µ ∈ X ∗ (T ): (1) µ is dominant integral (µτ,1 ≥ · · · ≥ µτ,2n for all ∀τ : F → E), (2) µ is a pure (∃w ∈ Z such that µτ,i + µτ,2n−i+1 = w, 1 ≤ j ≤ 2n, ∀τ : F → E), (3) Crit(µ) is nonempty. Later, when we study Manin relations, and especially for the application to nonvanishing of twists of L-functions, we will need to assume that Crit(µ) has at least two integers. 4.3. A degree-zero class. Recall from (34), that we have reached Hc• (S˜LHβ , V(j,w−j),O ), but we would like to work with integral cohomology with constant coefficients, i.e., with Hc• (S˜LHβ , O). Towards this, let’s note that the map det(h1 ) , det(h2 ) diag(h1 , h2 ) 7→ det(h2 ) gives us an identification of the connected components of S˜LHβ : (35)
π0 (S˜LHβ ) −→ C(pβ ) × C(det2 (Lβ )),
where the target is the product of ray-class groups of F modulo pβ , and modulo det2 (Lβ ) which stands for determinants from the second copy of GLn in Lβ . We emphasize that by equation (1) in Lemma 2.2 (i), det2 (Lβ ) and hence also C(det2 (Lβ )) is independent of the multi-exponent β. Consider an algebraic Hecke character η = η0 k k−w F , where η0 is a character of finite order that comes from a character η0 : det2 (Lβ ) → C× . (Such a character η will later play a role when we discuss Shalika models for representations of GL2n .) Define an automorphic character of H as:
det(h1 ) j
η(det(h2 ))−1
θη,j (diag(h1 , h2 )) := det(h2 ) This gives a global section ϑη,j for the sheaf V(−j, j−w),O on S˜LHβ , i.e., we have (36)
ϑη,j ∈ H 0 (S˜LHβ ; V(−j, j−w),O ).
(Here we assume that E is large enough so that O receives the values of η0 .) Taking the cup-product with ϑη,j gives us an isomorphism (37)
− ∪ ϑη,j : Hc• (S˜LHβ ; V(j,w−j),O ) −→ Hc• (S˜LHβ ; O),
and, as before, we will consider this isomorphism especially in degree • = q0 .
20
MLADEN DIMITROV,
FABIAN JANUSZEWSKI,
A. RAGHURAM
4.4. Summing over the second component. Now we specialize to cohomoology in degree • = q0 , and recall that q0 is the dimension of the manifold S˜LHβ . Recall also from (35) the identification π0 (S˜H ) with C(pβ ) × C(det2 (Lβ )). Let x ∈ C(pβ ) and y ∈ C(det2 (Lβ )), Lβ
and we think of x and y as finite ideles. Put d(x, y) := diag(xy, 1, . . . , 1, y, 1, . . . 1) ∈ H(A(∞) ),
(38)
which is the diagonal matrix with xy in the (1, 1)-th entry and y in the (n+1, n+1)-th entry, and 1’s everywhere else on the diagonal. Let S˜LHβ ,(x,y) denote the connected component of ] on the connected manifold S˜H indexed by the pair (x, y). Fix an orientation [S˜H Lβ
Lβ ,(x,y)
S˜LHβ ,(x,y) as (x, y) varies over C(pβ ) × C(det2 (Lβ )). These orientations are to be fixed in a consistent manner, which means that we fix an ordered basis, for once and for all, on ◦ Z ◦ ∩ H(R))◦ . Now we define a map the tangent space of the symmetric space H(R)◦ /(K∞ ∞ given by summing over the variable y: Sβ2 : Hcq0 (S˜LHβ , O) −→ O[C(pβ )],
(39) defined as
Sβ2 (ς)(x)
:=
X
y∈C(det2 (Lβ ))
for any ς ∈
Hcq0 (S˜LHβ , O)
and x ∈
C(pβ );
Z
ς
H [S˜L
β ,(x,y)
]
the integral is given by Poincar´e duality.
4.5. The distribution µφj,η . β . The evaluation map takes as input an element of 4.5.1. The evaluation map Ej,η q0 ˜H ∗ Hc (SLβ , ι Vµ,O ) and gives an O-valued function on C(pβ ) which is defined by the following diagram:
(40)
G; V Hcq0 (SK µ,O )
τβ,O ι∗β
// Hcq0 (S˜H ; ι∗ Vµ,O ) Lβ
κ∗j
// Hcq0 (S˜H ; V(j,w−j),O ) Lβ ❯❯❯❯ ❯❯❯❯ ❯❯❯❯ −∪ϑη,j ❯❯❯❯ ❯❯❯❯ ❯❯❯❯ ❯❯❯❯ Hcq0 (S˜LHβ ; O) ❯❯❯❯ β Ej,η ❯❯❯❯ ❯❯❯❯ ❯❯❯❯ Sβ 2 ❯❯❯❯ ❯**
O[C(pβ )]
4.5.2. Ordinarity and the Up -operator. For a multi-exponent β = (βp )p|p we introduce the notation Upβ := Upβ := [Ktβp K]. Then Upβ acts on p-integral cohomology via the •-action. A simultaneous eigenclass φ for Upβ , as β varies over any non-negative multiexponent, is called Q-ordinary of level K, if Y βp × Upβ • φ = αβp · φ, αβp = αp , and αp ∈ Zp . p|p
p-ADIC L-FUNCTIONS AND NONVANISHING
21
(In Sect. 5.2 we will take up a more detailed discussion of ordinarity.) This is equivalent to saying that φ is an eigenvector for the unnormalized action of Upβ with respective eigenvalue αβ0 = µ∨ (tβp )−1 · αβp , i.e., Q-ordinarity of φ is equivalent to (41)
|αβp |p = |(µ∨ )(tβp ) · αβ0 |p = 1.
∀β :
In fact, it suffices to veryify (41) for a single β satisfying βp > 0 for all p | p. G; V ord for the maximal O-submodule of H q0 (S G ; V We write Hcq0 (SK c µ,O ) µ,O ) on which K β the operators Up act invertibly via the •-action. 4.5.3. The distributive property. For any q | p, let 1q := (δpq )p|p denote the nonnegative tuple with 1 at the q-component and 0 in the other components. Theorem 4.1. Let the notations be as in (40). For β = (βp )p|p with βp ≥ 1 for all p | p, let for any q | p be πβ+1q : C(pβ+1q ) → C(pβ ) the canonical map. Then the following diagram commutes: β+1q
Ej,η
G; V Hcq0 (SK µ,O )
// O[C(pβ+1q )]
1
πβ+1q
Up q •(−) β Ej,η
G; V Hcq0 (SK µ,O )
// O[C(pβ )]
G; V Proof. Let φ ∈ Hcq0 (SK µ,O ) and fix a β with its components satisfying βp ≥ 1. Then 1q 1q β β ∗ ∗ ∗ Ej,η (Up • φ) = S2 κj (τβ,O ιβ (Up • φ)) ∪ ϑη,j X 1 = Sβ2 (κ∗j (ι∗ (µ∨ (tβp )τK,ξtβ t∗K,ξtβ µ∨ (tpq )τK,ut1q t∗ 1q φ)) ∪ ϑη,j ) p
= Sβ2 (κ∗j (ι∗
(42)
X
β+1q
µ∨ (tp
K,utp
p
p
u∈U (O/q)
)τK,ξtβ ut1q t∗ p
1
q K,ξtβ p utp
p
u∈U (O/q)
(φ)) ∪ ϑη,j )
Let pr : C(pβ+1q ) → C(pβ ) be the canonical map. Let x ∈ C(pβ ) and fix any x0 ∈ pr−1 (x). Then by (2) and the compatibility of Sβ2 with restriction, we obtain from (42) 1
(C(pβ ) : C(pβ+1q )) · (U (pβ ) : U (pβ+1q )) X β+1 β+1 µ∨ (tp q )τK,ξtβ ut1q t∗ · S2 q (κ∗j (ι∗
β Ej,η (Up q • φ)(x) =
p
u∈U (O/q)
By Lemma 2.1, to u is attached the element ku satisfying τK,ξtβ ut1q t∗ p
p
1
q K,ξtβ p utp
= τK,diag(k−1 ,1 u
β+1q n )ξtp
t∗
p
1
q K,ξtβ p utp
(φ)) ∪ ϑη,j )(x0 ). β+1q
−1 K,diag(ku,1 ,1n )ξtp
.
Furthermore, when u runs trough U (O/q), the element x0 ·det(ku ) ∈ C(pβ+1q ) runs through all preimages y of x under pr, each occurring with the same multiplicity. Therefore, by
22
MLADEN DIMITROV,
FABIAN JANUSZEWSKI,
A. RAGHURAM
1
β (Up q • φ)(x) can be written as Lemma 2.2, the expression for Ej,η
(C(pβ ) : C(pβ+1q )) · (U (pβ ) : U (pβ+1q )) X β+1 β+1 ·S2 q (κ∗j (ι∗ µ∨ (tp q )τK,diag(k−1 ,1 u
u∈U (O/q)
β+1q n )ξtp
t∗
β+1q
−1 K,diag(ku ,1n )ξtp
This expression simplifies to (C(pβ ) : C(pβ+1q )) · (U (pβ ) : U (pβ+1q )) X X
β+1 S2 q (κ∗j (ι∗
y∈pr−1 (x) u∈U (O/q) x0 det(ku )=y
µ
∨
(φ))∪ϑη,j )(x0 ).
β+1 (tp q )τK,ξtβ+1q t∗ β+1q (φ) K,ξtp p
) ∪ ϑη,j )(y),
which in turn may be written as X
β+1q
S2
∗ ι∗ (φ)) ∪ ϑη,j )(y) = (κ∗j (τβ+1 q ,O β+1q
y∈pr−1 (x)
X
β+1q
Ej,η
β+1q
(φ)(y) = πβ+1q (Ej,η
(φ)).
y∈pr−1 (x)
Here we exploited that the multiplicities cancel the index by (3); we also used that η is unramified at p.
G; V ord and 4.5.4. Definition of µφj,η (x). For an ordinary cohomology class φ ∈ Hcq0 (SK µ,O ) a class x ∈ C(pβ ) (∀p | p : βp ≥ 1) we put β (Up−β • φ)(x), µφj,η (x) := Ej,η
where Up−β denotes the inverse of Upβ as an operator on ordinary cohomology via the •-action. By Theorem 4.1, this is a well defined element in µφj,η ∈ O[[CF (p∞ )]] := lim O[C(pβ )]. ←− β
Here, CF (p∞ ) denotes the class group of F of level p∞ , i.e., β 0 b(p) CF (p∞ ) := lim F × \A× F /F∞ (OF )(1 + p OFp ). ←− β
4.6. Manin relations.
p-ADIC L-FUNCTIONS AND NONVANISHING
23
4.6.1. The main result on Manin relations. The norm map from F to Q induces a homomorphism NF/Q : CF (p∞ ) → CQ (p∞ ). We have CQ (p∞ ) ∼ = Z× p , giving us the p-adic cyc ∞ × cyclotomic character χQ : CQ (p ) → Zp . Composing by the norm map from F to Q gives us the p-adic cyclotomic character of F : χcyc : CF (p∞ ) → Z× p. F
(43)
Theorem 4.2 (Manin Relations). Let µ ∈ X0∗ (T ) be a pure dominant integral weight and G; V ord be an ordinary cohomology class. Whenever j, j + 1 ∈ Crit(µ), we let φ ∈ Hcq0 (SK µ,O ) have an equality of distributions on CF (p∞ ): φ φ χcyc F · µj,η = µj+1,η .
Before embarking on the proof of this theorem, we begin with some technical preparation. 4.6.2. Lie theoretic considerations. By the distributive property (cf. Theorem 4.1) we may reduce to strict pβ -power level with integral exponents β ∈ N, ignoring the finer components p | p for simplicity of notation. For this purpose we consider the diagonal matrix tp = diag(p1n , 1n ). Recall that b = t ⊕ n and q = h ⊕ u. We observe for any β ≥ 0 the relations tβp nO t−β ⊆ nO , tβp uO t−β = pβ uO . p p wn . A subscript ξ (−) denotes conjugation by ξ, for Also, recall the matrix ξ = 01nn w n − −1 − ξ example: bO = ξbO ξ . Proposition 4.3. We have the relations gO = hO + ξ b− O,
(44) and (45)
ξ
ξ − − (nO ∩ hO ) ⊆ [h, h]O + ξ n− O = [h, h]O + [qO , qO ].
Proof. Since ξ ∈ G(O), it suffices to verify (44) over E, where it boils down to show that dimE hE ∩ ξ b− E = n. To this end, let l1 , l2 be lower triangular matrices in Mn (E) and u ∈ Mn (E). Then l1 l1 + wn u l2wn − l1 − wn u −1 ξ· ·ξ = u l2 wn u l2wn − wn u
lies in hE if and only if u = 0 and l1 = l2wn . Therefore, l1 and l2 are diagonal matrices determining each other uniquely. This proves (44). Conjugating (45) by ξ −1 yields to the problem of solving n1 n1 h1 (h1 − h2 )wn = + n n2 n2 hw 2
for given diag(n1 , n2 ) ∈ h ∩ nO and unkown diag(h1 , h2 ) ∈ [hO , hO ] and diag(n1 , n2 ) ∈ n− . Evidently the choice n h1 = h2 = n1 + nw 2 ,
is a solution with the desired properties.
n n1 = −nw 2 ,
n n2 = −nw 1 ,
24
MLADEN DIMITROV,
FABIAN JANUSZEWSKI,
A. RAGHURAM
Corollary 4.4. For any β ≥ 0, inside U (gO ), we have the relations U (gO ) = U (hO ) · U (ξ b− O ),
(46) and β
βξ − bO ). U (ξtp nO ) ⊆ U ([h, h]O + pβ hO ) · U (ξ n− O+p
(47)
Proof. Relation (46) is a consequence of (44) and the Poincar´e-Birkhoff-Witt Theorem. For (47), the decomposition nO = (hO ∩ nO ) ⊕ uO , gives tβp nO t−β = (hO ∩ nO ) ⊕ pβ uO . p Conjugating by ξ we get −1 ξtβp nO t−β = ξ (hO ∩ nO ) ⊕ pβ ξ uO . p ξ
Apply the decomposition (45) to the first summand and (44) to the second to get −1 β βξ − ξ − + ξtβp nO t−β ξ ⊆ [h, h] + p h + p b n O O p O . O
Then (47) follows again by the Poincar´e-Birkhoff-Witt Theorem, because the sums within the parentheses on the right hand side are O-Lie algebras. 4.6.3. Lattices and the projection formula. Recall from (11) the lowest weight vector v0 ∈ Vµ,E and the G(O)-lattice Vµ,O = U (gO )·v0 = U (nO )·v0 . Recall also that the matrix tp acts on v0 via the scalar −µ∨ (t), where µ∨ denotes the highest weight of the contragredient representation of Vµ,E . The normalized action of tp on Vµ,E is given by tp • v := µ∨ (tp ) · tp v,
v ∈ Vµ,E .
Then tp • v0 = v0 and tp • Vµ,O ⊆ Vµ,O . Suppose j ∈ Crit(µ), and recall from Sect. 4.2 the map κj : Vµ,O → V(j,w−j),O . By (46) we have Vµ,O = U (hO ) · ξv0 , which implies κj (ξv0 ) 6= 0, and furthermore that every element in the image of Vµ,O under κj is an integral multiple of κj (ξv0 ), i.e., we get a map (48)
ιj : Vµ,O → O,
defined by
κj (v) = ιj (v)κj (ξv0 ).
It is independent from the choice of κj because of (33). We now come to the main technical theorem that is at the heart of the Manin relations. Theorem 4.5. For any β ≥ 0, and any v ∈ ξ · tβp • Vµ,O ⊂ Vµ,O
the element ιj (v) ∈ O/pβ O is independent of j ∈ Crit(µ), i.e., for j1 , j2 ∈ Crit(µ) we have (49)
ιj1 (v) ≡ ιj2 (v)
(mod pβ O).
Proof. For v ∈ ξ · tβp • Vµ,O , there exists m ∈ U (nO ) with v = ξ · tβp • (mv0 ) =
ξtβ p
β
m · ξv0 ∈ U (ξtp nO ) · ξv0 .
By Corollary 4.4, (47), write ξtp
m = xy,
βξ − x ∈ U ([h, h]O + pβ h), y ∈ U (ξ n− O + p bO ).
p-ADIC L-FUNCTIONS AND NONVANISHING
25
Let x0 , y0 ∈ O be the degree zero terms of x and y, respectively, and let x1 := x − x0 ∈ ([h, h]O + pβ h) · U ([h, h]O + pβ h), βξ − ξ − y1 := y − y0 ∈ (ξ nnO + pβ ξ b− O ) · U ( nO + p bO ),
be the higher degree terms in their respective enveloping algebras. Then ιj (v) = ιj (xy · ξv0 ) = ιj ((x0 + x1 ) · (y0 + y1 ) · ξv0 ) = (x0 + x1 ) · ιj ((y0 + y1 ) · ξv0 ) (since ιj is H-equivariant) ≡ x0 · ιj (y0 · ξv0 ) (mod pβ O) = x0 y0 · ιj (ξv0 ) = x0 y0 ,
(since x1 ∈ [h, h]O acts trivially on a line)
which does not depend on j as claimed.
4.6.4. Proof of Theorem 4.2. For any β ≥ 1 we consider the canonical projection πβ : O −→ O/pβ O. Recall the diagram (40). By construction, the map τβ,O : ι∗β Vµ,O −→ ι∗ Vµ,O , defined in (29), factors over the monomorphism i : ι∗ ρµ (ξ) tβp • Vµ,O −→ ι∗ Vµ,O
as in the diagram (20). Therefore, we have a canonical factorization τβ,O ιβ : Vµ,O −→ ι∗ ρµ (ξ) tβp • Vµ,O −→ ι∗ Vµ,O .
Now relation (49) from Theorem 4.5 translates to the statement that the composite ^ β O, πβ ιj τβ,O ιβ : Vµ,O −→ O/p is independent of j ∈ Crit(µ) when restricted to the connected component of the identity. On the other components, these maps differ only by the different actions of diag(x, 1, . . . , 1) on V(j,w−j),O/pβ O for varying j ∈ Crit(µ). These relations descend to cohomology and the diagram τβ,O ι∗β
κ∗j
// Hcq0 (S˜H ; V (j,w−j),O/pβ O ) Lβ ❯❯❯❯ ❯❯❯❯ ❯❯❯❯ −∪ϑη,j ❯❯❯❯ ❯❯❯❯ ❯❯❯❯ ❯❯❯❯ Hcq0 (S˜LHβ ; O/pβ O) ❯❯❯❯ β ❯❯❯❯ Ej,η ❯❯❯❯ ❯❯❯❯ Sβ ❯❯❯❯ 2 ❯**
G; V Hcq0 (SK µ,O )
// Hcq0 (S˜H ; ι∗ Vµ,O ) Lβ
(O/pβ O)[C(pβ )]
shows that by the definition of µφj,η and the action of diag(x, 1, . . . , 1) on V(j,w−j) we have NF/Q (x) · µφj,η (x) ≡ µφj+1,η (x) whenever j, j + 1 ∈ Crit(µ) and x ∈ C(pβ ).
(mod pβ ),
26
MLADEN DIMITROV,
FABIAN JANUSZEWSKI,
A. RAGHURAM
5. L-functions for GL(2n) 5.1. Representations with a Shalika model. This sub-section contains a brief review of the necessary ingredients from [GRa2], together with some complements such as multiplicity one results for Shalika models and a discussion involving refined Betti–Shalika periods. Henceforth, Π will stand for a cuspidal automorphic representation of G(A) = GL2n (AF ); as in [GRa2], such a Π need not be unitary. Keeping multiplicity one for GL2n in mind, we will let Π also stand for it’s representation space VΠ within the space of cusp forms for G(A). 5.1.1. Global Shalika models. Let ( ) h 0 1 X h ∈ GLn S := s = ⊂ GL2n 0 h 0 1 X ∈ Mn
× be a Hecke character such be the Shalika subgroup of GL2n . Let η : F × \A× F → C n that η equals the central character ωΠ of Π. Fix, once and for all, a nontrivial character ψ : F \AF → C× . We get an automorphic character: h 0 1 X × η ⊗ ψ : S(F )\S(AF ) → C , s = 7→ η(det(h))ψ(T r(X)). 0 h 0 1
For a cusp form ϕ ∈ Π and g ∈ G(A) consider the integral Z η η,ψ ϕ(sg)(η ⊗ ψ)−1 (s)ds, Sϕ (g) = Sϕ (g) := Z(A)S(F )\S(AF )
where ds is a right invariant Haar measure on S. It is well-defined by the cuspidality of the function ϕ (cf. [JacS, 8.1]) and hence yields a function Sϕη : G(A) → C which satisfies the transformation law Sϕη (sg) = (η ⊗ ψ)(s) · Sϕη (g), for all g ∈ G(A) and s ∈ S(A). In particular, we obtain an intertwining of G(A)-modules G(A)
Π → IndS(A) (η ⊗ ψ),
ϕ 7→ Sϕη .
The following theorem, due to Jacquet and Shalika, gives a necessary and sufficient condition for Sψη being non-zero. Theorem 5.1 ([JacS] Thm. 1, p. 213). The following assertions are equivalent: (i) There is a ϕ ∈ Π such that Sϕη 6= 0. G(A) (ii) Sψη defines an injection of G(A)-modules Π ֒→ IndS(A) (η ⊗ ψ). (iii) The twisted partial exterior square L-function Y LS (s, Π, ∧2 ⊗ η −1 ) := L(s, Πv , ∧2 ⊗ ηv−1 ) v∈S / Π
has a pole at s = 1, where SΠ is the set of places where Π is ramified.
This is proved in [JacS] for unitary representations and its extension to the non-unitary case is easy. If Π satisfies any one, and hence all, of the equivalent conditions of Theorem 5.1, then we say that Π has an (η, ψ)-Shalika model, and we call the isomorphic image Sψη (Π) of Π under Sψη a global (η, ψ)-Shalika model of Π. The following proposition gives another equivalent condition for Π to have a global Shalika model using the Asgari–Shahidi transfer [AS] from GSpin2n+1 to GL2n . See [GRa2] for more details.
p-ADIC L-FUNCTIONS AND NONVANISHING
27
Proposition 5.2. Let Π be a cuspidal automorphic representation of GL2n (AF ) with central character ωΠ . Then the following assertions are equivalent: (i) Π has a global (η, ψ)-Shalika model for some character η satisfying η n = ωΠ . (ii) Π is the transfer of a globally generic cuspidal automorphic representation π of GSpin2n+1 (AF ). In particular, if any of the above equivalent conditions is satisfied, then Π is essentially self-dual. The character η may be taken to be the central character of π. 5.1.2. Period integrals and L-functions. The following proposition, due to Friedberg and Jacquet, is crucial for much that will follow. It relates the period-integral over H of a cusp form ϕ of G to a certain zeta-integral of the function Sϕη in the Shalika model corresponding to ϕ over one copy of GLn . The measures are as in [GRa2, 2.8]. Proposition 5.3. [FJac, Prop. 2.3] Assume that Π has an (η, ψ)-Shalika model. For a cusp form ϕ ∈ Π, consider the integral s−1/2 Z h1 0 det(h1 ) η −1 (det(h2 )) d(h1 , h2 ). ϕ Ψ(s, ϕ) := 0 h2 det(h2 ) Z(A)H(Q)\H(A) Then Ψ(s, ϕ) converges absolutely for all s ∈ C. Next, consider the integral Z h 0 η Sϕ ζ(s, ϕ) := | det(h)|s−1/2 dh. 0 1 GLn (AF )
Then ζ(s, ϕ) is absolutely convergent for ℜ(s) ≫ 0. Further, for ℜ(s) ≫ 0, we have ζ(s, ϕ) = Ψ(s, ϕ), which provides an analytic continuation of ζ(s, ϕ) by setting ζ(s, ϕ) = Ψ(s, ϕ) for all s ∈ C. Suppose the representation Π of G(A) = GL2n (AF ) decomposes as Π = ⊗′v Πv , where Πv is an irreducible admissible representation of GL2n (Fv ). We have the local Definition 5.4. For any place v we say that Πv has a local (ηv , ψv )-Shalika model if there is a non-trivial (and hence injective) intertwining GL
(Fv )
Πv ֒→ IndS(F2n v)
(ηv ⊗ ψv ).
If Π has a global Shalika model, then Sψη defines local Shalika models at every place. The corresponding local intertwining operators are denoted by Sψηvv and their images by Sψηvv (Πv ), whence Sψη (Π) = ⊗′v Sψηvv (Πv ). We can now consider cusp forms ϕ such that the function ξϕ = Sϕη ∈ Sψη (Π) is factorizable as ξϕ = ⊗′v ξϕv , where GL
(Fv )
ξϕv ∈ Sψηvv (Πv ) ⊂ IndS(F2n v) Proposition 5.3 implies that Z ζv (s, ξϕv ) :=
GLn (Fv )
ξ ϕv
g1,v 0 0 1v
(ηv ⊗ ψv ).
| det(g1,v )|s−1/2 dg1,v
is absolutely convergent for ℜ(s) ≫ 0. The same remark applies to Z Y g1,f 0 ξ ϕf ζf (s, ξϕf ) := | det(g1,f )|s−1/2 dg1,f = ζv (s, ξϕv ). 0 1f GLn (A∞ ) F
v∈S / ∞
28
MLADEN DIMITROV,
FABIAN JANUSZEWSKI,
A. RAGHURAM
The next proposition, also due to Friedberg–Jacquet, relates the “Shalika-zeta-integral” with the standard L-function of Π. Proposition 5.5. [FJac, Propositions 3.1, 3.2] Assume that Π has an (η, ψ)-Shalika model. Then for each place v and ξϕv ∈ Sψηvv (Πv ) there is a holomorphic function P (s, ξϕv ) such that ζv (s, ξϕv ) = L(s, Πv )P (s, ξϕv ). One may hence analytically continue ζv (s, ξϕv ) by re-defining it to be L(s, Πv )P (s, ξϕv ) for all s ∈ C. Moreover, for every s ∈ C there exists a vector ξϕv ∈ Sψηvv (Πv ) such that P (s, ξϕv ) = 1. If v ∈ / SΠ , then this vector can be taken to be the spherical vector ηv ξΠv ∈ Sψv (Πv ) normalized by the condition ξΠv (idv ) = 1. Proposition 5.5 relates the Shalika-zeta-integral to L-functions, on the other hand Proposition 5.3 relates this integral to a period integral over H. The main thrust of [AG], refined and generalized in [GRa2], is that the period integral over H admits a cohomological interpretation, provided the cuspidal representation Π is of cohomological type. 5.1.3. Multiplicity one for Shalika models. Let’s record a result on the uniqueness of local (η, ψ)-Shalika model when η is possibly a nontrivial character. In the literature proofs may be found when η is trivial. See Nien [Ni] or Chen–Sun [ChS]. Their proofs go through mutatis mutandis while keeping track of the additional baggage of η. We simply record this as the following Proposition 5.6. Let π be an irreducible admissible representation of GL2n (F ) for a local field F . Suppose that there is character η of F × such that η n = ωπ the central character of π, and let ψ be a nontrivial character of F + . Then dim(HomS(F ) (π, η ⊗ ψ)) ≤ 1. This is also true if π is a representation that is fully induced from a character of the Borel subgroup. 5.1.4. An interlude on cuspidal cohomology. In this paragraph we recall some wellknown facts from Clozel [Cl, 3]. (See also [GRa2, 3.4] and [GRa1, 5.5].) Assume from now on that the cuspidal automorphic representation Π is cohomological, i.e., that there is a highest weight representation Vµ = ⊗τ ∈S∞ Vµτ of G∞ such that ◦ ◦ H q (g∞ , K∞ ; Π ⊗ Vµ,C ) = H q (g∞ , K∞ ; Π∞ ⊗ Vµ,C ) ⊗ Πf 6= 0
for some degree q. We will write this as Π ∈ Coh(G, µ). A necessary condition for the non-vanishing of this cohomology group is that the weight µ is pure. The archimedean component Π∞ can be explicitly described. For each archimedean place τ ∈ S∞ let ℓτ,i := µτ,i − µτ,2n−i+1 + 2(n − i) + 1 = 2µτ,i + 2(n − i) + 1 − w,
1 ≤ i ≤ n,
so ℓτ,1 > ℓτ,2 > · · · > ℓτ,n ≥ 1. For any integer ℓ ≥ 1 consider the unitary discrete series representation D(ℓ) of GL2 (R) of lowest non-negative SO2 -type ℓ + 1 and central character sgnℓ+1 . (Note that this normalization of the discrete series representation is as for example in [GRa2, 3.4]. It is not Q as in [Di, Definition 2.1].) Let R be the parabolic subgroup of GL2n with Levi factor ni=1 GL2 . Then GL2n (R) ∀τ ∈ S∞ , D(ℓτ,1 )| det |−w/2 ⊗ · · · ⊗ D(ℓτ,n )| det |−w/2 , Πτ ∼ = IndR(R)
p-ADIC L-FUNCTIONS AND NONVANISHING
29
◦ Z◦ ; Π using which H q (g∞ , K∞ ∞ ⊗ Vµ,C ) can be explicitly computed. The group of con∞ ◦ acts on this cohomology space. For any character ǫ of K /K ◦ nected components K∞ /K∞ ∞ ∞ ◦ )∗ , the ǫ-eigenspace in the top which we write as ǫ = (ǫτ )τ ∈S∞ ∈ (Z/2Z)S∞ ∼ = (K∞ /K∞ degree, i.e., in degree q0 := d(n2 + n − 1) is one-dimensional. Observe that q0 only depends on n and the degree d of F/Q. For a representation Π as above, it’s relative Lie algebra cohomology is a summand of cuspidal cohomology which in turn injects into cohomology with compact supports:
(50)
G K ◦ ◦ q G K ). ) ֒→ Hcq (SK ; Vµ,C H q (g∞ , K∞ Z∞ ; Π ⊗ Vµ,C ) ֒→ Hcusp (SK ; Vµ,C
(See, for example, the discussion in [GaRa, Sect. 2].) As proved in [Cl] cuspidal cohomolgy inherits a rational structure from sheaf-cohomology which allows one to deduce that the finite part Πf of such a cohomological cuspidal automorphic representation Π is defined over its rationality field that we denote Q(Π). 5.1.5. Shalika model versus cohomology. The reader should appreciate that the two conditions we may impose on Π: (i) the analytic condition of admitting an (η, ψ)-Shalika model, or (ii) the algebraic condition of contributing to the cuspidal cohomology of G, are in fact two entirely different conditions. One may construct examples of representations satisfying only one of these conditions and not the other; see [GRa2, Sect. 3.5]. Suppose now that Π is a cohomological cuspidal representation admitting an (η, ψ)- Shalika model, then Π∞ being cohomological forces its central character to be ωΠ∞ = Nm−nw . Furthermore, η is forced to be algebraic of the form η = η0 k k−w F for some character η0 of finite order; see the proof of Thm. 5.3 in [GaRa]. 5.1.6. Betti-Shalika periods. For a cuspidal representation Π which is cohomological ◦ , let’s recall an isomorphism that with respect to µ, and for every character ǫ of K∞ /K∞ identifies the Shalika-model of the finite part Πf with a summand in cuspidal cohomology: ◦ ; S η∞ (Π ) ⊗ V Fix a generator [Π∞ ]ǫ of the one-dimensional space H q0 (g∞ , K∞ ∞ µ,C )[ǫ]; this ψ∞ ǫ then fixes an isomorphism Θ of G(Af )-modules (exactly as in [RaS, 3.3] and [GRa2, Sect. 4.2]) as the composition of the three isomorphisms: η
∼
η
◦ ; Sψη∞ (Π∞ ) ⊗ Vµ,C )[ǫ] Sψff (Πf ) −→ Sψff (Πf ) ⊗ H q0 (g∞ , K∞ ∞ ∼
◦ −→ H q0 (g∞ , K∞ ; Sψη (Π) ⊗ Vµ,C )[ǫ] ∼
◦ −→ H q0 (g∞ , K∞ ; Π ⊗ Vµ,C )[ǫ],
where the first map is ξϕf 7→ ξϕf ⊗ [Π∞ ]ǫ ; the second map is the obvious one; and the third map is the map induced in cohomology by (Sψη )−1 . For any extension E/Q(Π), the Shalika η model Sψff (Πf ) has an E-rational structure, and independently, the cohomological model ǫ × ◦ ;Π⊗V H q0 (g∞ , K∞ µ,C )[ǫ] has an E-structure. The Betti-Shalika period ω (Π) ∈ C is the ǫ homothety that is needed to modify Θ so that (51)
Θǫ0 := ω ǫ (Π)−1 · Θǫ
preserves E-structures. See [GRa2, Defn./Prop. 4.2.1]. The periods are well-defined in C× /E × . (Actually, one defines a family of periods for all Galois conjugates of Π and this family is well-defined in (E ⊗ C)× /E × .) Later, we will refine these periods when E is a suitably large p-adic field to be well-defined up to units in E.
30
MLADEN DIMITROV,
FABIAN JANUSZEWSKI,
A. RAGHURAM
5.1.7. Algebraicity results for critical values. The main result of [GRa2] is the following: let χ be a finite-order Hecke character of F , and Crit(Π) = Crit(Π ⊗ χ) be the set of critical points in 21 + Z for the L-function L(s, Π ⊗ χ) of Π ⊗ χ. Let 21 + j ∈ Crit(Π ⊗ χ). Then, the quantity (52)
ω
L( 12 + j, Πf ⊗ χf ) ∈ Q, n χ (Π ) ω(Π ∞ , j) G(χf ) f
(−1)j ǫ
and is Gal(Q/Q)-equivariant. (The quantity ω(Π∞ , j) is a period at infinity, defined in [GRa2, Thm. 6.6.2], and which depends on a nonvanishing result of Sun [Su]; and G(χ) is the Gauss sum of χ.) The Betti–Shalika period ω ǫ (Π) depends on the choice [Π∞ ]ǫ , j however, the product ω (−1) ǫχ (Πf ) ω(Π∞ , j) in the denominator of the above expression is independent of this choice. It is proved in [Jan4, Thm. A] that we may choose our class [Π∞ ]ǫ such that for all 12 + j ∈ Crit(Π ⊗ χ) we have ω(Π∞ , j) = (2πi)jdn , where, recall that d = [F : Q]. Hence, we may refine the quantity in (52) as Lalg ( 12
(53)
+ j, Πf ⊗ χf ) :=
L( 21 + j, Πf ⊗ χf ) jǫ
ω (−1)
χ
(Πf ) G(χf )n (2πi)jdn
∈ Q.
We aim at p-adically interpolating Lalg ( 12 +j, Πf ⊗χf ) as χ varies over characters of p-power conductor. 5.2. Local aspects: p-ordinarity and nonvanishing of a local zeta integral. 5.2.1. p-ordinary representations. Fix a cuspidal representation Π of G(A) that is cohomological with respect to a highest weight µ. Assume that the local component Πp , for I each p | p, has vectors fixed by the Iwahori subgroup, i.e., Πp p 6= 0. Hence, we have an unramified algebraic character λp : T2n (Fp ) → C× such that Πp occurs as a subquotient in
GL
GL (F ) e G IB (λp ) := a IndB2n2n(Fp )p λ p,
(F )
where a IndB2n2n(Fp )p denotes unnormalized or algebraic induction and
We remark that a
ep : λ
diag(t1 , . . . , t2n ) 7→
2n Y
|ti |2n−i λp,i (ti ).
i=1
2n−1 2n−1 2n−1 GL (F ) e GL2n IndB2n2n(Fp )p λ p = IndB2n (Fp ) (λp,1 | | 2 ⊗ λp,2 | | 2 ⊗ · · · ⊗ λp,2n | | 2 ).
the right hand side is the normalized parabolic induction. Fix a finite extension E/Qp with valuation ring O ⊆ E together with an embedding ip : Q(Π) → E. Recall that the field of rationality Q(Π) of Π, by definition, comes with a canonical embedding iC : Q(Π) → C. The character λp being algebriac means that we may assume, after enlarging E if necessary, G (λ ) is an admissible representation of GL (F ) over that λp takes values in E × . Then IB p 2n p E. For an admissible representation (π, V ) of GL2n (Fp ) over E, the Jacquet module with respect to the Borel subgroup is defined as VB := V /h{π(u)v − v | u ∈ U2n (Fp ), v ∈ V }i.
p-ADIC L-FUNCTIONS AND NONVANISHING
31
This is an admissible representation of T2n (Fp ). Let’s introduce the matrices tp,ν := diag(̟p 1ν , 12n−ν ) ∈ GL2n (Fp ) where ̟p is a uniformizer for p, and consider the Hecke operators Up,ν := [Ip tp,ν Ip ],
1 ≤ ν ≤ 2n.
Let qp denotes the cardinality of the residue field of Fp . We have the well-known Proposition 5.7 (Prop. 5.4 and Cor. 5.5 in [Hi2]). With the notations as above, assume that the characters λωp , as ω varies over the Weyl group W (GL2n , T2n ), are pairwise distinct. G (λ ) is a semisimple T (F )-module: Then the Jacquet module of IB p 2n p M G fωp . λ IB (λp ) B = ω∈W (GL2n ,T2n )
fωp -isotypic component in the right hand side is a simultaneous eigenEach vector v in the λ vector of Up,ν , 1 ≤ ν ≤ 2n, and − ν(ν−1) 2
(54)
Up,ν v = qp
· λωp (tp,ν ) · v.
We will also need the following Proposition 5.8. With the notations as above, assume that the characters λωp , as ω I
varies in W (GL2n , T2n ), are pairwise distinct. Suppose Wp ∈ Πp p is a simultaneous Up,ν -eigenvector, 1 ≤ ν < 2n, with non-zero eigenvalues then there exists a unique ω ∈ W (GL2n , T2n ) such that (55)
−ν(ν−1) 2
Up,ν Wp = qp
· λωp (tp,ν ) · Wp ,
for all 1 ≤ ν ≤ 2n.
Furthermore, Wp lies in a unique line in Πp characterized by (55). Proof. For an admissible representation (π, V ) of B2n (Fp ) over E, let’s recall that the functor V → V U2n (O) of taking invariants under U2n (O) is exact, and we have a surjection U
V U2n (O) ։ VB 2n
(56)
(O)
.
Equation (5.4) on page 678 of [Hi2] gives (57)
U
Πp 2n
(O)
U
= (Πp )B2n
(O)
U (O) U (O) , → (Πp )B2n ⊕ ker Πp 2n
and each Up,ν acts nilpotently on the second summand on the right hand side. Therefore, U (O) by our hypothesis on Wp , this vector maps to a non-zero Up,ν -eigenvector Wp,B ∈ (Πp )B2n with same eigenvalue. By our hypothesis on λp , we get that (54) implies (55). This also implies the uniqueness of Wp up to a scalar. Recall the maximal (n, n)-parabolic subgroup Q ⊆ G. We call the representation Π Q-ordinary at p of Iwahoric level (resp., B-ordinary at p of Iwahoric level) if it is Qordinary (resp., B-ordinary) in the sense of [Hi1, Hi2]. Let’s recall these notions. Since Π G , and use this realization is cohomological, we may realize each Πp in the cohomology of SK to transfer the •-action for the operators Up,ν to Πp over E. With this convention Π is I B-ordinary at p if and only if we find for every p | p a non-zero v ∈ Πp p satisfying (58)
Up,ν • v = αp,ν · v,
ip (αp,ν ) ∈ O× ,
for all 1 ≤ ν ≤ 2n.
32
MLADEN DIMITROV,
FABIAN JANUSZEWSKI,
A. RAGHURAM
We say Π is Q-ordinary, if Up,n • v = αp,n · v, ip (αp,n ) ∈ O× . Clearly, B-ordinarity implies Q-ordinarity, but the converse is false in general. By Corollary 8.3 (and its proof) in [Hi2], B-ordinarity of Π implies that Πp occurs in an G (λ ) with an algebraic character λ satisfying the hypotheses of induced representation IB p p Prop. 5.8 for all p | p, namely that λωp for ω ∈ W (G2n , T2n ) are all distinct. Furthermore, the p-adic unit eigenvalue αp is uniquely determined. In particular, the B-ordinary component I of Πp p is free of rank 1. Let’s make condition (58) explicit in terms of the Satake parameters. Recall that the cohomological weight µ of Π is a rational character µ : T = ResF/Q T2n → GL1 which is defined over its field of rationality Q(µ) ⊆ Q(Π). Therefore, µ∨ induces a character Y ∨ T (Qp ) = T2n (Fp ) → E × . µ∨ p = ⊗p|p µp : p|p
Then Up,ν • Wp = µ∨ p (tp,ν ) · Up,ν Wp . ∨ The local components µ∨ p of µp may
be themselves be considered as 2n-tuples of char-
acters ResFp /Qp GL1 → GL1 , 1 ≤ i ≤ 2n, µ∨ p,i : subject to the corresponding ResFp /Qp B2n -dominance condition. When Wp is a simultaneous eigenvector for all these Hecke operators, by Prop. 5.8, after reordering the components of λp if necessary, gives −
Up,ν • Wp = qp
ν(ν−1) 2
· µ∨ p (tp,ν ) · λp (tp,ν ) · Wp .
Therefore, if | · |E denotes the norm on E, then Wp is a B-ordinary vector if and only if −ν(ν−1) ∨ = 1, for all 1 ≤ ν ≤ 2n. qp 2 · µ (t ) · λ (t ) (59) p p,ν p p,ν E
It’s easy to see that (59) is equivalent to 1−ν ∨ q (60) · µ (̟p ) · λp,ν (̟p ) p
p,ν
E
= 1,
for all 1 ≤ ν ≤ 2n.
5.2.2. p-ordinarity for representations with a Shalika model. Now let’s suppose that the cuspidal representation Π of G(A) that is cohomological with respect to µ also admits an (η, ψ)-Shalika model Sψη (Π), together with our standing assumption that Πp is Iwahori-spherical for all p diving a fixed rational prime p. We may and shall assume that Πp is a quotient GL
(F )
IndB2n2n(Fp )p (λp,1 | |
2n−1 2
⊗ · · · ⊗ λp,2n | |
2n−1 2
) ։ Πp .
A Shalika functional for Πp gives a Shalika functional for the induced representation, which gives us a symmetry condition ([AG, Prop.1.3]) λp,2n−i+1 | |
2n−1 2
= (λp,i | |
2n−1 2
)−1 ηp ,
1 ≤ i ≤ n.
This symmetry may also be written as λp,2n−i+1 (̟p ) = qp2n−1 λp,i (̟p )−1 (ηp (̟p ). (As an example to keep in mind, the reader should think of the Steinberg representation of GL2n , which admits a Shalika model for ηp being the trivial character, and which appears GL
(F )
as a quotient of IndB2n2n(Fp )p (| |−
(2n−1) 2
⊗| |−
(2n−3) 2
⊗· · ·⊗| |
2n−1 2
).) We also have the symmetry
p-ADIC L-FUNCTIONS AND NONVANISHING
33
coming from the purity condition on µ, namely, that if w is the purity weight of µ (which is independent of i and p | p), we may attach to w the character wp := NFwp /Qp :
ResFp /Qp GL1 → GL1 .
Then, in additive notation, µp,2n−i+1 = wp − µp,i ,
or, equivalently,
∨ µ∨ p,2n−i+1 = −wp − µp,i .
It is easily seen that (59) for 1 ≤ ν ≤ 2n is equivalent to the conditions 1−i ∨ q · µp,i (̟p ) · λp,i (̟p ) E = 1, 1 ≤ i ≤ n, and p (61) wp (̟p )−1 · ηp (̟) = 1. E
The first of these conditions specializes, when F = Q and µ trivial, exactly to the condition of ordinarity in [AG, (A1), Sect. 2.1]. 5.2.3. Non-vanishing of the local zeta integral. We revert to local notation as in Sect. 2. Recall the local zeta integral in Thm. 2.3: 1
ζ(s; W, χ) = c · G(χ)n · N(fχ )n(s− 2 )−
n(n+1) 2
· W (diag(1n , wn )),
where W is an Iwahori-spherical Shalika function on GL2n (F ). In applications, it’s crucial to know that W (diag(1n , wn )) 6= 0. We now solve this problem with a careful choice of W when the ambient representation of GL2n (F ) containing W is spherical. Let K2n = GL2n (O), the standard maximal compact subgroup of GL2n (F ), and define the parahoric subgroup of type (n, n) in GL2n (F ) as a b (62) J2n := k = ∈ K2n | a, c ∈ GLn (O), c ∈ Mn (p), d ∈ Mn (O) , c d
i.e., it is the subgroup of GL2n (O) which is block upper-triangular mod p. Clearly, we have I2n ⊂ J2n ⊂ K2n . Suppose W is a Shalika vector that is fixed by J2n then it is also fixed by I2n and Thm. 2.3 is applicable; however, W (diag(1n , wn )) 6= 0 ⇐⇒ W (12n ) 6= 0, since diag(1n , wn ) ∈ J2n . As in Sect. 5.2.2, incorporating the symmetry condition, let’s write π as a quotient: GL
(F )
−1 K : Ind(χ) := IndB2n2n(F ) (χ1 ⊗ · · · ⊗ χn ⊗ χ−1 n η ⊗ · · · ⊗ χ1 η) ։ π. (2n−1)
(So, χj is λp,j | | 2 from Sect. 5.2.2.) Let αj = χj (̟) and θ = η(̟), then the Satake −1 parameter of Ind(χ) is diag(α1 , . . . , αn , α−1 n θ, . . . , α1 θ). From the induced representation we can map into its Shalika model using an explicit integral (see [AG, (1.3)]). However, we need to account for our symmetry condition being written differently. Consider the following diagram: (63)
GL
(F )
−1 IndB2n2n(F ) (χ1 ⊗ · · · ⊗ χn ⊗ χ−1 1 η ⊗ · · · ⊗ χn η)
❱❱❱❱ ❱❱❱❱ SI ❱❱❱❱ T ❱❱❱❱ ❱❱❱❱ ❱❱❱❱ ❱❱❱++ GL2n (F ) −1 −1 IndB2n (F ) (χ1 ⊗ · · · ⊗ χn ⊗ χn η ⊗ · · · ⊗ χ1 η) ❣❣33 C ❣ ❣ ❣ ❣❣❣ ❣ ❣ ❣ ❣ ❣❣❣❣❣ K ❣❣❣❣❣Sπ ❣ ❣ ❣ ❣ ❣ ❣❣❣❣❣ π ❣❣
34
MLADEN DIMITROV,
FABIAN JANUSZEWSKI,
A. RAGHURAM
where, T is the standard intertwining operator between the two induced representations, Sπ an (η, ψ) Shalika functional on π, and SI an (η, ψ)-Shalika functional that is given by the integral in [AG, (1.3)] which we recall: for f ∈ Ind(χ), define (64) Z Z 1n 1n X h ¯ f SI (f )(g) := g η −1 (det(h))ψ(tr(X)) dX dh 1n 1n h Bn \Gn Mn
The integral is well-defined, i.e., moving h ∈ Gn to bh for b ∈ Bn , leaves the integrand unchanged. By [AG, Lem. 1.5], this integral converges in a certain domain, and admits an analytic continuation after multiplying by Y (65) P (χ) := (1 − αi αj θ −1 ). 1≤i w2 and the corresponding critical s = 21 +j then does not lie inside k−j χ) 6= 0. Therefore, the interior of the critical strip for L(s, Π), whence L( 21 + j, Π ⊗ ωQ k cyc (ωQ µΠ )|C (p∞ ) 6= 0 as claimed. Q
Proof of Thm. 5.15. By the Weierstrass Peparation Theorem, a non-zero power series in one variable over O admits only finitely many zeroes. This means that there are at most finitely many characters χ : CQcyc (p∞ ) → C× with Z χdµΠ = 0, cyc ∞ CQ (p )
because this integration corresponds to evaluation of the univariate power series corresponding to µΠ |C cyc (p∞ ) . Therefore, by Theorem 5.13, L( w+1 2 , Π ⊗ χ) 6= 0 for all but Q finitely many χ as claimed. 5.4.2. Variations-I: A stronger nonvanishing result. The same argument gives a slightly stronger nonvanishing result which we formulate as follows: Theorem 5.17. Under the hypotheses of Theorem 5.15, let ν be a character of CF (p∞ ) of finite order. Then for all but finitely many Hecke characters χ of Q of finite order and with p-power conductor we have: L( w+1 2 , Π ⊗ νχ) 6= 0.
p-ADIC L-FUNCTIONS AND NONVANISHING
39
k ν ′ for some integer k and ν ′ a character of C cyc (p∞ ). Therefore, Proof. Observe that ν = ωQ Q kµ the restriction of νµΠ to CQcyc (p∞ ) agrees with the ν ′ -translate of the restriction of ωQ Π cyc ∞ to CQ (p ), which is non-zero by Theorem 5.16. The rest of the proof proceeds mutatis mutandis as the proof of of Theorem 5.15.
This result is slightly stronger because the representation Π ⊗ ν, even though of cohomological type and admiting a Shalika model, is still not in the same class as Π because the local representations at p are no longer unramified (or Iwahori spherical). 5.4.3. Variations-II: simultaneous nonvanishing. The following corollary of Thm. 5.15 follows from the fact that we have nonvanishing for all but finitely many Hecke characters χ of Q of finite order and with p-power conductor. Corollary 5.18. Let n1 , . . . , nr be positive integers. For 1 ≤ k ≤ r, let µk ∈ X0∗ (T2nk ) be a pure dominant integral weight for GL(2nk ) over F . Suppose that each µk satisfies the regularity condition in (69) and all the purity weights wk := w(µk ) are even integers. Let Πk ∈ Coh(GL2nk , µk ) be a representation with a nontrivial (ηk , ψ)-Shalika model. For a rational prime p, suppose that each Πk is unramified at p, and is B-ordinary at p of Iwahoric level. Then, for all but finitely many Hecke characters χ of Q of finite order and with p-power conductor, we have: L( w12+1 , Π1 ⊗ χ)L( w22+1 , Π2 ⊗ χ) · · · L( wr2+1 , Πr ⊗ χ) 6= 0. Let’s note that this is a simultaneous nonvanishing result at s = 1/2 for certain unitary cuspidal representations: L( 12 , Πu1 ⊗ χ)L( 12 , Πu2 ⊗ χ) · · · L( 21 , Πur ⊗ χ) 6= 0. We will leave it to the reader to formulate the stronger version of simultaneous nonvanishing using Thm. 5.17 and Cor. 5.18. 5.4.4. Variations-III: critical values of the symmetric 5-th power L-functions. As an example illustrating an application of simultaneous nonvanishing to algebraicity results, let’s consider ∆ ∈ S12 (SL2 (Z)) the Ramanujan ∆-function, and let π(∆) be the associated cuspidal automorphic representation of GL2 (A). A particular case of Cor. 5.18 gives infinitely many twisting characters χ such that (71)
L( 12 , Sym3 (π(∆)) ⊗ χ)L( 12 , π(∆) ⊗ χ) 6= 0.
Now using Cor. 5.2 and Prop. 5.4 of [Ra], we get an algebraicity result for the central critical value (72)
L( 12 , Sym5 (π(∆)) ⊗ ξ).
working exactly analogous to [Ra, (5.6)]; we leave the whimsical details to the interested reader. In the classical normalization this L-value is the same as L(28, Sym5 ⊗ξ, ∆). References [AS] [A] [AG] [BR]
M. Asgari, F. Shahidi, Functoriality for General Spin Groups, arXiv:1101.3467v1 preprint (2011) A. Ash, Non-square-integrable cohomology classes of arithmetic groups, Duke J. Math. 47 (1980) pp. 435–449 A. Ash and D. Ginzburg, p-adic L-functions for GL(2n), Invent. Math. 116 (1994), 27-73. L. Barthel and D. Ramakrishnan, A non-vanishing result for twists of L-functions for GL(n), Duke Mathematical Journal, Vol 74, No. 3 (1994) 681–700.
40
[Cl]
[ChS] [CFH]
[De]
[Di] [FJac] [Ge] [GaRa]
[GRa1] [GRa2] [Hi1] [Hi2]
[JacS]
[Jan1] [Jan2] [Jan3]
[Jan4] [L] [Ni] [RaS] [Ra]
[Ro] [Sh] [Sp] [Su] [VZ]
MLADEN DIMITROV,
FABIAN JANUSZEWSKI,
A. RAGHURAM
L. Clozel, Motifs et formes automorphes: applications du principe de fonctorialit´e. Automorphic forms, Shimura varieties, and L-functions, Vol. I (Ann Arbor, MI, 1988), 77-159, Perspect. Math., 10, Academic Press, Boston, MA, 1990. F. Chen and B. Sun, Uniqueness of twisted linear periods and twisted Shalika periods, Preprint available at https://arxiv.org/abs/1703.06238 G. Chinta, S. Friedberg and J. Hoffstein, Asymptotics for sums of twisted L-functions and applications. In Automorphic representations, L-functions and applications: progress and prospects, 75–94, Ohio State Univ. Math. Res. Inst. Publ., 11, de Gruyter, Berlin, 2005. P. Deligne, Valeurs de fonctions L et p´eriodes d’int´egrales, Automorphic forms, representations and L-functions (Providence RI) (A. Borel and W. Casselman, eds.), Proceedings of the Symposium in Pure Mathematics 33(2), American Mathematical Society, 1979, pp. 313–346. M. Dimitrov, Automorphic symbols, p-adic L-functions and ordinary cohomology of Hilbert modular varieties, Amer. J. Math., 135 (2013), pp. 1–39. S. Friedberg, H. Jacquet, Linear periods, J. reine angew. Math. 443 (1993) pp. 91–139. L. Gehrmann, Shalika models and p-adic L-functions. To appear in Israel J. Math. W.T. Gan and A. Raghuram, Arithmeticity for periods of automorphic forms, In “Automorphic Representations and L-functions,” 187–229, Tata Inst. Fundam. Res. Stud. Math., 22, Tata Inst. Fund. Res., Mumbai, 2013. H. Grobner, A. Raghuram, On some arithmetic properties of automorphic forms of GLm over a division algebra, Int. J. Number Theory, Vol. 10, No. 4 (2014) 963–1013. H. Grobner and A. Raghuram, On the arithmetic of Shalika models and the critical values of Lfunctions for GL(2n), Amer. J. Math., 136 (2014) 675–728. H. Hida, Control Theorems of p-nearly ordinary cohomology groups for SL(n), Bull. Soc. Math. France, 123 (1995) 425–475. H. Hida. Automorphic induction and Leopoldt type conjectures for GL(n), Asian Journal of Mathematics 2 (1998), special issue ‘Mikio Sato: A great Japanese Mathematician of the Twentieth Century’, 667–710. H. Jacquet, J. Shalika, Exterior square L-functions, in: Automorphic forms, Shimura varieties, and L-functions, Vol. II, Perspect. Math., vol. 10, eds. L. Clozel and J. S. Milne, (Ann Arbor, MI, 1988) Academic Press, Boston, MA, 1990, pp. 143–226. F. Januszewski, Modular Symbols for reductive groups and p-adic Rankin-Selberg convolutions over number fields, J. reine angew. Math. 653 (2011), doi:10.1515/CRELLE.2011.019, page 1–45. F. Januszewski, On p-adic L-functions for GL(n) × GL(n − 1) over totally real fields, International Mathematics Research Notices 2015 no. 17 (2015), pp. 7884–7949, doi:10.1093/imrn/rnu181. F. Januszewski, p-adic L-functions for Rankin-Selberg convolutions over number fields, Annales mathmatiques du Qu´ebec (2016) 40, special issue in honor of Glenn Steven’s 60th Birthday, pp. 453–489. F. Januszewski, On Period Relations for Automorphic L-Functions II, preprint available at http://arxiv.org/abs/1504.06973. W. Luo, non-vanishing of L-functions for GLn (AQ ). Duke Math. J. 128 (2005), no. 2, 199–207. C. Nien, Uniqueness of Shalika models. Canad. J. Math. 61 (2009), no. 6, 1325–1340. A. Raghuram, F. Shahidi, On certain period relations for cusp forms on GLn , Int. Math. Res. Not. Vol. (2008), doi:10.1093/imrn/rnn077. A. Raghuram, On the special values of certain Rankin-Selberg L-functions and applications to odd symmetric power L-functions of modular forms, Int. Math. Res. Not., Vol. (2010) 334–372, doi:10.1093/imrn/rnp127. D. Rohrlich, Nonvanishing of L-functions for GL(2). Invent. Math. 97 (1989), no. 2, 381–403. G. Shimura, On the periods of modular forms. Math. Ann., 229, no. 3, (1977) 211–221. B. Speh, Unitary representations of GL(n, R) with non-trivial (g, K)-cohomology, Invent. Math. 71 (1983), 443-465. B. Sun, Cohomologically induced distinguished representations and a non-vanishing hypothesis for algebraicity of critical L-values, Preprint available at arXiv:1111.2636 D. Vogan and G. Zuckerman, Unitary representations with non-zero cohomology, Compositio Math. 53 (1984), 51–90.
p-ADIC L-FUNCTIONS AND NONVANISHING
41
´ Lille 1, UMR CNRS 8524, UFR Math´ Mladen Dimitrov: Universite emathiques, 59655, Villeneuve d’Ascq Cedex, FRANCE E-mail address:
[email protected] URL: http://math.univ-lille1.fr/~mladen/ ¨ r Algebra und Geometrie, Fakulta ¨ t fu ¨ r Mathematik, Fabian Januszewski: Institute fu ¨ r Technologie (KIT), GERMANY Karlsruher Institut fu E-mail address:
[email protected] URL: http://www.math.kit.edu/iag3/~januszewski/en A. Raghuram: Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pashan, Pune 411021, INDIA. Current address: Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA. E-mail address:
[email protected],
[email protected] URL: https://sites.google.com/site/math4raghuram/home
