SOME REMARKS ON K-LATTICES AND THE ADELIC HEISENBERG GROUP FOR CM CURVES
arXiv:1407.3351v1 [math.NT] 12 Jul 2014
FRANCESCO D’ANDREA AND DAVIDE FRANCO
Abstract. We define an adelic version of a CM curve which takes care of all the automorphisms and which is naturally equipped with an embedding into the adelic Heisenberg group. We embed into the adelic Heisenberg group the set of commensurability classes of arithmetic 1-dimensional K-lattices and define theta functions on it. We thus get an algebra of functions on the set of 1-dimensional K-lattices which is equipped with an action of the Heisenberg group and which exibits a nice behavior under complex automorphisms (Theorem 7.12).
1. Introduction After the seminal paper [2], many efforts have been devoted in recent years to the construction of quantum systems incorporating explicit class field theory for a imaginary quadratic number field K ([6], [9], [8], [4], [7]). More specifically, in [6] a quantum statistical mechanical system fully incorporating the explicit class field theory is exibited in terms of commensurability of 1-dimensional K-lattices. The connection between class field theory and quantum statistical mechanics is provided by a C ∗ -dynamical system containing an arithmetic subalgebra AQ with symmetry group isomorphic to Gal(Kab , K) and a set of fabulous states sending AQ to Kab , in such a way that the symmetry group action is compatible with Galois’ one. This arithmetic subalgebra is defined by means of the modular field, namely the field of modular function defined over Qab . This paper stems from an attempt to define an algebra of function on commensurability classes of (arithmetic) 1-dimensional K-lattices, with a nice behavior under Galois action, by means of adelic theta functions and adelic Heisenberg group ([14]). Motivated by a purely algebraic definition of adelic theta function over an abelian variety and their deformations, David Mumford introduced in a celebrated series of papers of the sixties ([13]) the finite Heisenberg group acting on the sections of an ample line bundle defined on the abelian variety. This led him to an adelic version of any abelian variety, defined as an extension of the set of its torsion points by the Barsotti-Tate group ([14], Ch. 3). It turned out that sections of the pull back of some line bundle on the tower of isogenies ([14], Definition 4.26), the so called adelic theta functions, are acted on by an adelic version of the Heisenberg group ([14], Ch.4). 2010 Mathematics Subject Classification. 11R56, 11R37, 11G15, 14K25, 58B34. Keywords. K-lattices, complex multiplication, adelic Heisenberg group, adelic theta functions. Acknowledgments. This research was partially supported by UniNA and Compagnia di San Paolo under the grant “STAR Program 2013”.
2
F. D’ANDREA AND D. FRANCO
In this work we apply Mumford’s constructions to elliptic curves with complex multiplication (CM curves for short) and compare it with the moduli spaces of 1dimensional K-lattices introduced in [6]. We define an adelic version of a CM curve which takes care of all the automorphisms and which is naturally equipped with an embedding into the adelic Heisenberg group. This allow us to implement the rich authomorphisms structure of the CM curve in the definitions of Heisenberg group and theta functions and to interpret by means of Class Field Theory (Corollary 7.8, Corollary 7.11) the usual nice behavior of theta functions under authomorpisms fixing the Hilbert field of K ([14], Proposition 5.6). One of our main results is an extension of the Main Theorem of Complex Multiplication ([16] II, Theorem 8.2) involving the Heisenberg group which allows us to give a complete description of the behavior of theta functions under complex authomorphisms (Theorem 7.7, Theorem 7.10). Finally, we embed into the adelic Heisenberg group the set of commensurability classes of arithmetic 1-dimensional K-lattices in order define theta functions on it. Then we apply our results to get an algebra of functions on the set of 1-dimensional K-lattices which is equipped with an action of the Heisenberg group and exibiting a nice behavior under complex automorphisms (Theorem 7.12). The paper is organized as follows. In Section 2 we recall some basic facts that will be needed in the sequel concerning limits of torsion groups obtained as quotients of lattices in a number field. We will be as self-contained as possible in order of both fixing notations and facilitating the reader. In Section 3 we introduce an adelic version of a CM curve and describe canonical parametrizations of its points. In Section 4 we recall the spaces of 1-dimensional K-lattices introduced in [6] and compare them with adelic CM curves in order to obtain natural morphisms into the Heisenberg group. In Section 5 we introduce the Adelic Heisenberg group of a CM curve and embed the adelic CM curve into it by means of a symmetric line bundle. We also describe the action of the complex automorphisms fixing the Hilbert field of K on the Heisenberg group. In Section 6 we state and prove a version of the Main Theorem of Complex Multiplication (see [16] II, Theorem 8.2) concerning Heisenberg groups. In a nutshell, what such a theorem says is that if two different CM curves are mapped into each other by a complex automorphism then the embeddings into their Heisenberg groups can be made coherent with such a map. Finally, in Section 7 we introduce theta functions and study their behavior under complex automorphisms. 2. Some basic facts Let K be a number field. Denote by R the ring of algebraic integers of K, by I = IR the set of (integral) ideals of R, by J = JR the group of fractional ideals of R, freely generated by the primes ([12], p. 91) and by Cl(R) the ideal class group of R. Set moreover A = AK,f the ring of finite ad`eles of K, and ˆ = lim R ⊂ A R ←− I I∈IR the completion of R. Recall that I −1 = {x ∈ K | xI ⊂ R} ∈ J
K-LATTICES AND THE ADELIC HEISENBERG GROUP FOR CM CURVES
is a fractional ideal s.t. I · I −1 = R . Then K : of R I −1 = R
I −1 R
can be identified with a submodule
K A xI = 0 . x∈ ' ˆ R R
Similarly, if Λ ⊂ K is a fractional ideal of K then K submodule of Λ : I −1 Λ = Λ
3
I −1 Λ Λ
can be identified with a
K A ˆ := Λ · R ˆ ⊂ A. xI = 0 , Λ x∈ ' ˆ Λ Λ −1
It is well known (see e.g. [16], II Proposition 1.4) that I Λ Λ is a free R I module −1 −1 of rank 1. If J ∈ I is another ideal s.t. I ⊂ J, then obviously I Λ Λ ⊃ J Λ Λ . Denote by S the set of non-archimedean places of K ([3], p. 189), choose a prime element πv , ∀v ∈ S ([3], p. 42) and set π : v → πv . Definition 2.1. Consider the multiplicative set Rπ ⊂ R containing all the elements r = πvn11 . . . πvnkk , and denote by Φπ the inverse limit of the following inverse system: −1 (a) Λ (a)−1 Λ · ab (b)−1 Λ , πab , Rπ , , if b | a. πab : −→ Λ Λ Λ An element u = (ua )a∈Rπ ∈ Φπ is said to be a coherent system of generators of (a)−1 Λ −1 R , πab , Rπ , if ua ∈ (a)Λ Λ is a (a) -module generator, ∀a ∈ Rπ . Λ The following Lemma shows that the choice made above gives rise to a canonical ˆ coherent system of generator which makes Φπ a free a R-module of rank 1: Lemma 2.2. There exists a canonical coherent system of generators whose limit ˆ uπ := lim ua ∈ Φπ is a generator of Φπ as a free R-module of rank 1. ←− Proof. Set a = Πv∈S πva(v) , b = Πv∈S πvb(v) , Λ ' Πv∈S Pve(v) , and assume a = bc, in Rπ . Then we have (πv )e(v)−a(v) (b)−1 Λ (πv )e(v)−b(v) (a)−1 Λ ' Πv , ' Πv . e(v) Λ Λ (πv ) (πv )e(v) To conclude it suffices to consider the following commutative diagram:
a(v)−e(v)
(a)−1 Λ Λ
α
(b)−1 Λ Λ
β
b(v)−e(v)
R (a)
R (b)
where α = Πv πv , β = Πv πv . The thesis follows from [15], Corollary ˆ = lim R is compact 1.1.6, since the orizzontal maps are module isomorpisms and R ←− (a) ˆ in the isomorphism above). (uπ corresponds to 1 ∈ R
4
F. D’ANDREA AND D. FRANCO
ˆ Remark 2.3. The Lemma shows that there is a canonical R-module generator (a)−1 Λ uπ ∈ lim Λ (the limit of a canonical system of generators) for any map π. ←− In what follows refer to uπ as the canonical generator associated to π and we set u = uπ when no confusion arises. Consider now two ideals I, J ∈ IR , s.t. I ⊂ J, and two fractional ideals Λ, Λ0 . −1 0 −1 −1 0 −1 R Then both I Λ Λ and I Λ0Λ J Λ Λ and J Λ0Λ are R I -modules J -modules and −1 −1 0 −1 0 −1 −1 −1 one can look at Hom R I Λ Λ , I Λ0Λ Hom R J Λ Λ , J Λ0Λ . Since I Λ Λ ⊃ J Λ Λ I J −1 −1 −1 0 one can restrict to J Λ Λ any morphism φ ∈ Hom R I Λ Λ , I Λ0Λ . Since Jφ(x) = I −1 −1 Λ gives a well defined φ(Jx) = 0 in J Λ Λ , ∀ x ∈ J Λ Λ , then the restriction φ J −1 Λ J −1 Λ J −1 Λ0 and we have thus defined a morphism: element in Hom R Λ , Λ0 J
rIJ : Hom R I
I −1 Λ I −1 Λ0 , Λ Λ0
→ Hom R J
J −1 Λ J −1 Λ0 , Λ Λ0
.
Similarly we also have restriction morphisms: −1 K K I Λ I −1 Λ0 rI : HomR , → Hom R , . I Λ Λ0 Λ Λ0 −1 −1 0 The triples Hom R I Λ Λ , I Λ0Λ , rIJ , IR define an inverse system, infact a surI jective inverse system ([15], p. 9) as we will see shortly, whose inverse limit can be K K identified with HomR Λ , Λ0 : Proposition 2.4. lim Hom R I ←−
I∈IR
I −1 Λ I −1 Λ0 , Λ Λ0
' HomR
K K , Λ Λ0
Proof. Obviously rJ = rIJ ◦ rI so we have a morphism −1 I Λ I −1 Λ0 K K HomR , 0 → lim Hom R , I ←− Λ Λ Λ Λ0 which easily turns out to be injective. To prove surjectivity one can either invoke K K compactness of HomR Λ , Λ0 and recall [15], Corollary 1.1.6 or argue as follow. Consider an element −1 I Λ I −1 Λ0 (φI )I∈I ∈ lim Hom R , , I ←− Λ Λ0 −1
I Λ fix x ∈ K (x). In order to Λ and choose I ∈ I s.t. x ∈ Λ . Set φ(x) := φ I K K show that such a φ is a well defined element of HomR Λ , Λ0 we have to prove −1 that φI (x) does not depend on the ideal I s.t. x ∈ I Λ Λ . Assume that also J is −1
−1
Λ s.t. x ∈ J Λ Λ , then both I and J are contained in I +J and (I+J) ⊂ Λ so φI (x) = φJ (x) = φI+J (x) hence φ is well defined and we are done.
I −1 Λ J −1 Λ Λ ∩ Λ ,
Observe that the set {(a) : a ∈ Rπ } is a cofinal subset of IR , so combining Lemma 1.1.9 of [15] with the isomorphism above, we obtain: −1 −1 K K I Λ I −1 Λ0 (a) Λ (a)−1 Λ0 R , 0 ' lim Hom R , ' lim Hom , . HomR I ←− ←− (a) Λ Λ Λ Λ0 Λ Λ0 I∈I a∈Rπ
K-LATTICES AND THE ADELIC HEISENBERG GROUP FOR CM CURVES −1
Lemma 2.5. Any choice of coherent systems of generators ua ∈ (a)Λ −1 0 va ∈ (a)Λ0 Λ a∈Rπ provides an explicit isomorphism: −1 R (a) Λ (a)−1 Λ0 ˆ lim Hom R ' lim , ' R. ←− ←− (a) (a) Λ Λ0 a∈Rπ
Λ a∈Rπ
R (a)
giv-
R (a) ,
b | a, are the natural projections. −1 −1 0 Fix (φa )a∈Rπ ∈ lim Hom R (a)Λ Λ , (a)Λ0 Λ . Then φa is uniquely determined by ←− (a) R ia (φa ) ∈ (a) such that φa (ua ) = ia (φa )va and it is very easy to show that the map: −1 (a) Λ (a)−1 Λ0 R ia : Hom R , → , φa → ia (φa ) (a) Λ Λ0 (a)
where pab :
→
and
a∈Rπ
−1 −1 0 R Proof. We define (a) -module isomorphisms ia : Hom R (a)Λ Λ , (a)Λ0 Λ → (a) ing the components ([15], p. 5) of an isomorphism −1 (a) Λ (a)−1 Λ0 R , r , R → Θ : Hom R , , p , R ab π ab π , (a) Λ Λ0 (a)
R (b)
5
R -module isomorphism. In order to conclude the proof we need to prove the is a (a) compatibility of the components just defined which amounts to verify the commutativity of the following diagram, for any b | a:
Hom
R (a)
rab
(a)−1 Λ (a)−1 Λ0 , Λ0 Λ
Hom R
(b)
(b)−1 Λ (b)−1 Λ0 , Λ0 Λ
j
i pab
R (a)
R (b)
R where i := ia and j := ib . Set i(φa ) = α and j(φb ) = β, to show that α ≡ β in (b) it −1 0 suffices to prove that αvb = βvb ∈ (b) 0 Λ . Recalling that φb = φa (b)−1 Λ , ub = a ua Λ
βvb = φb (ub ) = φa (ub ) = φa
b
Λ
and vb = ab va , we have: a
a a ua = φa (ua ) = αva = αvb b b b
and we are done.
Corollary 2.6. Any choice of coherent systems of generators ua ∈ −1 0 and va ∈ (a)Λ0 Λ a∈Rπ provides an explicit isomorphism: K K ˆ , 0 ' R. HomR Λ Λ
−1
(a) Λ
Λ a∈Rπ
3. Adelic elliptic curves with complex multiplication. Assume now K is quadratic imaginary and let Λ ⊂ K be a fractional ideal of K. C Then E = EΛ = Λ is an elliptic curve with End(E) = R ([16], p. 99). Following [16] p. 102, for a ∈ R we denote by E(a) the (a)-torsion points of E: E(a) := {x ∈ E | ax = 0}. By [16], Proposition 1.4 E(a) ' section to conclude:
(a)−1 Λ Λ
and we may apply the results of previous
6
F. D’ANDREA AND D. FRANCO
ˆ module Proposition 3.1. The Barsotti-Tate module T (E) := lim E(a) is a free R ←− of rank 1. ˆ Remark 3.2. Observe that any coherent systems of generators represents a Rmodule generator of T (E). Now we imitate [14] Definition 4.1 to define an adelic version of E which takes into account its complex multiplication structure. Definition 3.3. Set a n o V (E) := (xa )a∈Rπ ∈ E Rπ xa = xb , if b | a, x1 ∈ Etor . b K Definition above implies that x1 ∈ Etor ' Λ and of course there is a projection π : V (E) → Etor , s.t. π(xa ) = x1 . Set A = AK,f the ring of finite ad`eles of K. Theorem 3.4. (1) Any morphism φ ∈ HomR
K K R, Λ
gives rise to map of short exact sequences:
ˆ R
0
A
ψ
0
ψ
T (E)
K R
0
K Λ
0
φ
V (E)
The morphism φ is an isomorphism iff ψ(1) ∈ T (E) is a generator. (2) Any choice of a coherent systems of generators u = ua ∈ ˆ gives rise to a commutative diagram of R-modules: 0
ˆ R
A
0
T (E)
V (E)
π
K R
0
Etor
0
Proof. (1) Denote by πa the natural projection πa : A →
A ˆ aR
'
K aR .
(a)−1 Λ Λ a∈Rπ
Then we set: π (α) a ψ : A → V (E), ψ(α) = (xa )a∈Rπ , xa = φ . a Since bxab = bφ πabab(α) = φ πaa(α) = xa then φ(α) ∈ V (E). Obviously the following diagram commutes: A
K R
ψ
V (E)
φ
Etor
We are left to check the commutativity of the following diagram: ˆ R u
T (E)
A ψ
V (E)
K-LATTICES AND THE ADELIC HEISENBERG GROUP FOR CM CURVES
7
ˆ Then we have where ψ(1) = u = (ua )a∈Rπ . Fix γ = (γa )a∈Rπ ∈ R. 1 γ πa (γ) a γu = (γa ua )a∈Rπ = γa φ = φ = φ = ψ(γ) a a∈Rπ a a a∈Rπ a∈Rπ and we are done. To conclude it suffice to observe that since K K ˆ → T (E) , Ker → ' Coker R R Λ then φ is invertible iff ψ(1) is a generator of T (E). K K , Λ sending to (2) By Corollary 2.6, there exists a unique element ψ ∈ HomR R −1 u the canonical coherent system of generators a1 ∈ (a)R R a∈R , so the statement π follows by (1). 4. The moduli space of K-lattices. Let us recall the following (see [6], Def. 4.1): Definition 4.1. (1) A 1-dimensional K lattice (Λ, φ) is a finitely generated R-submodule Λ ⊂ C K s.t. Λ ⊗R K ' K, together with a R-morphism φ : R → KΛ Λ . (2) An invertible 1-dimensional K lattice (Λ, φ) is a finitely generated R-submodule K → KΛ Λ ⊂ C s.t. Λ ⊗R K ' K, together with a R-isomorphism φ : R Λ . Since for any finitely generated R-submodule Λ ⊂ C there exists k ∈ C s.t. kΛ ⊂ K ([5], Lemma 3.111) we also give the following: Definition 4.2. An arithmetic (invertible) 1-dimensional K lattice (Λ, φ) is a finitely generated R-submodule Λ ⊂ K s.t. Λ⊗R K ' K, together with a R-morphism K → KΛ (isomorphism) φ : R Λ . The following result is an immediate consequence of Theorem 3.4. K K Theorem 4.3. Any morphism φ ∈ HomR R , Λ gives rise to map of short exact sequences: 0
ˆ R
A
K R
0
0
T (E)
V (E)
K Λ
0
The corresponding K-lattice is invertible iff the vertical maps are isomorphisms. In K K particular any φ ∈ HomR R , Λ corresponds to a point of xφ ∈ T (E) (the image of C ) and the corresponding K-lattice is invertible iff the canonical generator of T R xφ is a generator of the Tate group. Following [16], pag 98 we denote by ELL(R) the moduli space of elliptic curves with End(E) ' R. Then ([16], II Proposition 1.2) ELL(R) is a Cl(R)-torsor. Definition 4.4. Denote by L the set of K-lattices modulo dilations and by L∗ the set of invertible K-lattices modulo dilations. Fix ρ ∈ A∗ and [Λ] ∈ Cl(R). By abuse of notation we denote by ρ[Λ] the effect of the usual action of A∗ on Cl(R) and by Eρ[Λ] ∈ ELL(R) the corresponding elliptic curve. Then the multiplication by ρ K K provides a well defined invertible element of HomRˆ R , ρR hence a point ψρ ∈ L∗ .
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F. D’ANDREA AND D. FRANCO
Corollary 4.5. (1) The map: φ : A∗ → L∗ , ρ → ψρ is an isomorphism and the projection L∗ → Cl(R) coincides with the usual Class Field map ([10], p. 224). ˆ × ˆ ∗ L∗ ' R ˆ × ˆ ∗ A∗ . (2) L ' R R R Lemma 4.6. Consider an arithmetic 1-dimensional K lattice (Λ, φ). Any 1-dimensional K lattice which is commensurable to (Λ, φ) is arithmetic. Proof. Assume that (Λ0 , ψ) is commensurable to (Λ, φ). Then both the natural projections Λ → Λ ∩ Λ0 , Λ0 → Λ ∩ Λ0 have finite index hence the same happens for Λ + Λ0 → Λ ∩ Λ0 . So there exists an integer n s.t. n(Λ + Λ0 ) ⊂ Λ ∩ Λ0 . The thesis follows since nΛ0 ⊂ n(Λ + Λ0 ) ⊂ Λ ∩ Λ0 ⊂ K. Lemma 4.7. Consider two fractional ideals Λ = ΠP ∈P P e(P ) , Λ0 = ΠP ∈P P f (P ) K K K K , Λ → HomR R , Λ0 induced by the and assume Λ ⊂ Λ0 . Then the map HomR R projection is represented, with respect to canonical generators (recall Remark 2.3), ·ρ e(P )−f (P ) by the multiplication by ρ = ΠP ∈P πP : A → A. Proof. It suffices to prove the Lemma for each localization KP . Fix an ideal I = ΠP ∈P P g(P ) . Then we have e(P )−g(P )
πP
and
−1 0 IP ΛP Λ0P
'
−1 IP ΛP ΛP
e(P )−g(P )
'
f (P )−g(P ) (πP ) f (P ) (πP )
(πP
e(P )
(πP
'
)
RP g(P ) (πP )
)
'
RP g(P ) (πP )
with canonical generator f (P )−g(P )
with canonical generator πP
.
Then the canonical projection acts as e(P )−g(P )
πP
e(P )−g(P )
→ πP
e(P )−f (P )
= πP
f (P )−g(P )
· πP
and we are done.
∗
Notations 4.8. Fix a fractional ideal Λ. We denote by T (EΛ ) the ring of generC ators of the Tate group of EΛ := Λ . Corollary 4.9. (1) Two arithmetic invertible 1-dimensional K lattices are commensurable iff they coincide. (2) Any arithmetic 1-dimensional K lattice is commensurable to an invertible one. (3) The set of arithmetic 1-dimensional K lattices modulo commensurability is S ∗ Λ T (EΛ ) . Proof. (1) Assume that (Λ, φ) and (Λ, φ) are commensurable and invertible. Then ˆ ∗ . Projecting they are represented with respect to canonical generators by ρ, ρ0 ∈ R 0 0 0 modulo Λ + Λ and recalling Lemma 4.7, we have α · ρ = α · ρ where α and α0 are suitable products of prime elements. Since none of the divisors of α and α0 divides ρ and ρ0 we have α = α0 and ρ = ρ0 . (2) Any arithmetic lattice is represented with respect to canonical generators by ˆ which can be written as ρ = α · ρ0 where α is a suitable product an element ρ ∈ R ˆ ∗ . Then ρ0 ∈ R ˆ ∗ can be interpreted as an invertible of prime elements and ρ0 ∈ R lattice for some Λ0 ⊂ Λ. (3) It follows from (1) and (2).
K-LATTICES AND THE ADELIC HEISENBERG GROUP FOR CM CURVES
9
Definition 4.10. Consider two fractional ideals Λ and Λ0 and set Γ = Λ ∩ Λ0 . De K K K K K K K K note by p : HomR R , Λ → HomR R , Γ and by q : HomR R , Λ0 → HomR R ,Γ the natural projection and consider the corresponding ρ and ρ0 obtained by means of Lemma 4.7. Then the lattices (Λ, φ) and (Λ0 , φ0 ) are commensurable iff p(φ) = p0 (φ0 ) so that the set of such commensurable lattices can be identified with the fiber product FΛ,Λ0 ⊂ T (EΛ ) × T (EΛ0 ) arising from the cartesian square: FΛ,Λ0
T (EΛ )
q
p
T (EΛ0 )
T (EΓ )
Keep notations as in Definition 4.10, then Corollary 4.9 implies the following: Theorem 4.11. Fix a set of representatives Λi of Cl(R), 1 ≤ i ≤ ]Cl(R). Set E = {1, 2, . . . , ]Cl(R)}×J . Then the groupoid of commensurability modulo dilations can be identified with: [ [ FΛi ,Λ ⊂ T (EΛi ) × T (EΛ ). (Λi ,Λ)∈E
(Λi ,Λ)∈E
0
Proof. If Λ and Λ are commensurable, Lemma 4.6 implies that there exists k ∈ C s.t. kΛ = Λi and kΛ0 ∈ Cl(R). Then we may conclude by means of Corollary 4.9. 5. Adelic Heisenberg group. In this section we follow [14], Ch. 4 in order to define an Adelic Heisenberg Group for an elliptic CM curve E. We begin with the following (see [14], Definition 4.5): Definition 5.1. Fix a morphism of complex manifolds f : Y → X and line bundle L on X. Then f ∗ L denotes the fibre product: f ∗L
fˆ
q
L p
Y
f
X
with p := πL and q := πf ∗ L natural projections. Assume that φ0 : Y → Y , φ : X → X are isomorphisms such that f ◦ φ0 = φ ◦ f and consider an authomorphism g = (φ, ψ) ∈ Aut(L). Then f ◦ φ0 ◦ q = φ ◦ f ◦ q = φ ◦ p ◦ fˆ = p ◦ ψ ◦ fˆ hence ψ ◦ fˆ : f ∗ L → L and φ0 ◦ q : f ∗ L → Y can be lifted to a morphism ψ 0 : f ∗ L → f ∗ L in such a way that g 0 := l(g) = (φ0 , ψ 0 ) ∈ Aut(f ∗ L), l : Aut(L)φ → Aut(f ∗ L)φ0 . Observe that ψ 0 ◦ fˆ = fˆ ◦ ψ. We will say that g 0 covers g over φ0 and, by abuse of notations, we will set ψ = f (ψ 0 ). For any line bundle L over E '
C Λ
we define as usual
K(L) := {x ∈ E | t∗x L ' L},
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F. D’ANDREA AND D. FRANCO
where t∗x is the translation by x. Keep notations as in [1] Ch. 6 and denote by G(L) the theta group of L and by G(L) its canonical covering. Proposition 5.2. Fix a ∈ R. For any x ∈ K(L) and any y ∈ E s.t. ay = x, we have a canonical isomorphism G(L)x ' G(a∗ L)y lifting over the translation by y. Proof. Since tx ◦ a = a ◦ ty we can apply the construction of Definition 5.1 to f = a, φ = tx and φ0 = ty to get a morphism G(L)x ' Aut(L)tx → Aut(a∗ L)ty ' G(a∗ L)y . Since G(L)x ' G(a∗ L)y ' C∗ (see [1] Ch. 6), we are done.
Lemma 5.3. If x ∈ Etor then the set: Ix := {a ∈ R | ax ∈ K(L)} is an ideal of R. Proof. We keep notations as in [1], 2.2 and set L = L(H, χ), where H = c1 (L) is the hermitian form and χ is the semicharacter associated to L via Appell-Humbert Theorem. Then we have: Λ(L) , Λ(L) := {v ∈ C | Im H(v, Λ) ⊂ Z} K(L) ' Λ (see [1], p. 37). Assume that v ∈ C projects on x ∈ E and that av ∈ Λ(L). Then, for any r ∈ R we have Im H(rav, Λ) = Im H(av, rΛ) ⊂ Z. Note that since 0 ∈ K(L), Ix contains the annihilator of x, hence is non empty. Notations 5.4. By abuse of notations we also set Ix = Ix1 for any x ∈ V (E). Theorem 5.5. Fix x = (xa )a∈R ∈ V (E), then we have xa ∈ K(a∗ L), ∀a ∈ Ix . Proof. Claim: a∗ (tx L ⊗ L−1 ) ' tax L ⊗ L−1 , ∀a ∈ R, ∀x ∈ E and ∀L ∈ P ic(E). Indeed, if v ∈ C projects on x and L = L(H, χ) via Appell-Humbert Theorem, then (tx L⊗L−1 ) is represented in P ic0 (E) by the character e2πi Im H(v, · ) ([1], p. 33). But then Lemma 2.3.4 of [1] implies that a∗ (tx L ⊗ L−1 ) is represented by the character e2πi Im H(av, · ) and the claim follows. Finally we have: txa a∗ L ⊗ (a∗ L)−1 ' a∗ (taxa L ⊗ L−1 ) ' tax1 L ⊗ L−1 and tax1 L ⊗ L−1 is trivial since a ∈ Ix , so that xa ∈ K(a∗ L).
Notations 5.6. For any x ∈ V (E) we set Jx = {a ∈ R | xa ∈ K(a∗ L)}. Theorem 5.5 implies that Ix ⊂ Jx , hence Jx 6= ∅. ˆ Definition 5.7. We denote by G(L) the adelic Heisenberg group i.e. the set of α = (x, (φa )a∈Jx ) s.t. x ∈ V (E), φa ∈ G(a∗ L)xa with φab covering txab over φa , ∀a ∈ Jx . Any collection of elements (φa )a∈Jx , φa ∈ G(a∗ L)xa with φab covering txab over φa will be called a coherent system of authomorphism defined over Jx . ˆ Remark 5.8. By 5.1, if α = (x, (φa )a∈Jx ) ∈ G(L) then b(φab ) = φa , i.e. ˆb ◦ φab = ˆ φa ◦ b, ∀a ∈ Jx .
K-LATTICES AND THE ADELIC HEISENBERG GROUP FOR CM CURVES
11
Theorem 5.9. (1) We have an exact sequence: ˆ 1 → C∗ → G(L) → V (E) → 0. (2) ∗ ˆ G(L) |p−1 ' p−1 ∗ a G(a L) a (K(a L))
(3) ˆ G(L) |T (E) ' C, ˆ so there exists a morphism σ L : T (E) → G(L) providing a section of the restriction to T (E) of the sequence above (σ(x) is defined as the unique ˆ element of G(L) lifting the identity). ˆ Proof. (1) By Theorem 5.5 the map G(L) → V (E) is surjective because ∅ = 6 Ix ⊂ Jx , ∀x ∈ V (E). Furthermore, for any x ∈ V (E), Proposition 5.2 implies that any coherent system of authomorphisms defined over Jx , (φa )a∈Jx , φa ∈ G(a∗ L)xa , is uniquely determined by any fixed φa , so we have: ˆ x ' G(a∗ L)x ' C∗ , ∀a ∈ Jx . G(L) a ∗ (2) For any x ∈ p−1 a (K(a L)) we obviously have that a ∈ Jx . By the isomorphism above we have: ∗ ∗ ˆ ' p−1 G(L) |p−1 a (G(a L)xa ), ∀xa ∈ K(a L) a (xa )
and we are done. (3) For any x ∈ T (E) we have 1 ∈ Jx and the statement follows as above.
ˆ Remark 5.10. If y, y 0 ∈ G(L) are chosen in such a way that π(y) = x, π(y 0 ) = x0 , then we have a well defined skewsymmetric map e : V (E) × V (E) → C1 ,
e(x, x0 ) = yy 0 y −1 y 0−1 ,
with values in the group of roots of unity of C1 ([14], Proposition 4.14). Assume now that L is symmetric: (−1)∗ L ' L. Correspondingly we have an involution i : G(L) → G(L). By abuse of notations we use the same symbol for the ˆ corresponding morphism of G(L): ˆ ˆ i : G(L) → G(L). We recall the following ([14], p. 58): ˆ ˆ Definition 5.11. Fix x ∈ V (E), y ∈ G(L) s.t. 2π(y) = x. Then τ (x) ∈ G(L) −1 defined as τ (x) := yi(y) does not depend of the choice of y, so we have a map: ˆ τ : V (E) → G(L), providing a section of the exact sequence of Theorem 5.9. Proposition 5.12. τ (x)τ (y) = e( x2 , y)τ (x + y)
12
F. D’ANDREA AND D. FRANCO
ˆ ˆ Proof. Observe that, for any pair x0 ∈ G(L), y 0 ∈ G(L) s.t. π(x0 ) = x and π(y 0 ) = y ˆ ˆ we have x0 y 0 = e(x, y)y 0 x0 . Choose z ∈ G(L), w ∈ G(L) s.t. 2π(z) = x, 2π(w) = y. −1 −1 So we have 2π(i(z) ) = x and 2π(i(w) ) = y hence y τ (x)τ (y) = zi(z)−1 wi(w)−1 = e( x2 , )zwi(z)−1 i(w)−1 2 = e( x2 , y)zwi(w)−1 i(z)−1 = zwi(zw)−1 = e( x2 , y)τ (x + y). Assume now both E and L be defined in the Hilbert field of K and L is the line bundle providing the embedding E → P2 via Weierstrass equation. By Theorem 4.3, any σ ∈ Gal(H, H) comes from a map (still denoted by σ with an abuse of notation) V (E) → V (E) obtained by multiplication with an element of T (E)∗ . ˆ and we Proposition 5.13. Any σ ∈ Gal(H, H) acts as automorphism of G(L) ˆ have: τ (σ(x)) = σ(τ (x)) ∈ G(L), ∀x ∈ V (E). Proof. By [16], Theorem 2.2 (b), the multiplication map by any a ∈ R is still defined over H hence in our hypotheses the same happens for a∗ L. Since Λ(a∗ L) is a fractional ideal of K containing Λ and σ is represented by some element of R ∗ in Aut( Λ(aΛ L) ) ' K(a∗ L) ([1], p. 37), we have an action of σ on G(a∗ L), ∀a ∈ R. Moreover such actions are obviously compatible with multiplication maps by elements of R hence σ : V (E) → V (E) lifts to an automorphism ˆ ˆ σ : G(L) → G(L). In order to conclude the proof we have τ (σ(x)) = σ(y)i(σ(y))−1 for any y s.t. 2π(y) = x, since 2σ(π(y)) = σ(2π(y)) = σ(x) and τ does not depend on y 0 s.t. 2π(y)0 = σ(x). ˆ ˆ Moreover, since σ : G(L) → G(L) is a morphism commuting with i we have −1 −1 σ(y)i(σ(y)) = σ(yi(x) ) = σ(τ (x)). 6. The main Theorem of Complex Multiplication for Adelic Curves. Consider an elliptic curve with complex multiplication by R embedded into P2 by means of the usual Weierstrass equation. Recall the Main Theorem of Complex Multiplication ([16] II, Theorem 8.2, see also [17], Ch. 5 and [11], Ch. 10): Theorem 6.1. Let σ ∈ Aut(C) and let s be an idele of K corresponding to σ via Artin map. Fix a complex analytic isomorphism: C → E(C), Λ where Λ is a fractional ideal. Then there exists a unique complex analytic isomorphism: C g : −1 → E σ (C), s Λ so that the following diagram commutes: f:
K Λ
s−1
g
f
E(C)
K s−1 Λ
σ
E σ (C)
K-LATTICES AND THE ADELIC HEISENBERG GROUP FOR CM CURVES
13
The main purpose of this section is to lift such a commutative diagram to adelic Heisenberg groups. To ease notations we put E 0 = E σ . Theorem 6.2. The map above extends to a unique isomorphism: s−1 : V (E) → V (E 0 ). Such an isomorphism lifts to an isomorphism of adelic Heisenberg groups ˆ ˆ 0 ): commuting with the sections τ : V (E) → G(L) and τ 0 : V (E 0 ) → G(L (τ (x))σ = τ 0 (σ(x)),
∀x ∈ V (E).
Proof. By proposition 4.3, the map s−1 extends to a isomorphism of short-exactsequences: 0
T (E)
V (E)
K Λ
0
0
T (E 0 )
V (E 0 )
K s−1 Λ
0
Denote by L (L0 ) the very ample line bundle over E (E 0 ) providing the embedding of E (E 0 ) in P2 : f1 : E → P2 (f10 : E 0 → P2 ) and fix a ∈ R. The isogeny a : E → E C C can be lifted to the identity map C → C by setting a : E ' Λ → aΛ ' E ([1], 0 0 p. 151), providing in such a way a natural embedding H (E, L) ⊂ H (E, a∗ L). Thus, by choosing suitable bases in H 0 (E, a∗ L), we can construct isomorphism, fa : E → Ea ⊂ Pha (ha = dimH 0 (E, a∗ L), and rational projections πa : Pha − − → P2 so that f1 = πa ◦ fa . Put Ea0 = Eaσ . Since E σ = E 0 and since σ commutes with πa , we have that πa : Ea0 → E 0 is a morphism such that: Ea
πa
σ
Ea0
E σ
πa
E0
s−1 Λ Λ ' ' Ker(Ea → E), as−1 Λ aΛ so E 0 ' Ea0 and πa : Ea0 → E 0 is the multiplication by a. Furthermore, the commutativity of the following diagram: K aΛ
πa
σ K as−1 Λ
K Λ
σ πa
K s−1 Λ
K K −1 implies that the isomorphism σ : aΛ → as−1 Λ is provided by multiplication by s and we have the commutativity of any square such as: K abΛ
b
σ K abs−1 Λ
K aΛ
σ b
K as−1 Λ
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F. D’ANDREA AND D. FRANCO
hence σ commutes with the multiplication map by any b ∈ R. For any automorphism φ ∈ G(a∗ L) we have an automorphism φσ = σ ◦ φ ◦ σ −1 ∈ G(a∗ L0 ) and a group isomorphism σ : G(a∗ L) ↔ G(a∗ L0 ). If φ is canonically represented by some element U ∈ P Gl(H 0 (E, a∗ L)) ([1], 6.4), by the choices made above φσ is canonically represented by U σ ∈ P Gl(H 0 (E 0 , a∗ L0 )). By the last commutative square b(φσab ) = (b(φab ))σ = φσa as soon as b(φab ) = φa , so σ sends a coherent system of automorphisms for E to a coherent system of ˆ automorphisms for E 0 and we have the asserted isomorphism between G(L) and 0 ˆ ): G(L ˆ ˆ 0 ). σ : G(L) → G(L Keeping notations as above, we finally have (τ (x))σ = σ ◦ τ (x) ◦ σ −1 = σ ◦ yi(y)−1 ◦ σ −1 = y σ i(y σ ) = τ 0 (σ(x)), because 2π(y σ ) = xσ as soon as 2π(y) = x. 7. Adelic Thetas By abuse of notations, for any (tx , ψ) ∈ G(a∗ L) we still denote by ψ its image via canonical representation ([1], 6.4): ρ˜a : G(a∗ L) → GL(H 0 (a∗ L)), ψ = ρ˜a (ψ) : s → ψ ◦ s ◦ t−x . Lemma 7.1. Consider ψ ∈ G((ab)∗ L) and φ ∈ G(a∗ L) s.t. b(ψ) = φ. Then we have a commutative square: H 0 (a∗ L)
b∗
φ
H 0 ((ab)∗ L) ψ
H 0 (a∗ L)
b∗
H 0 ((ab)∗ L)
Proof. For any s ∈ H 0 (a∗ L), b∗ (s) ∈ H 0 ((ab)∗ L) is characterized by ˆb ◦ b∗ (s) = s ◦ b so we are left to prove that ˆb ◦ ψ(b∗ (s)) = φ(s) ◦ b, ∀s ∈ H 0 (a∗ L). We have ˆb ◦ ψ(b∗ (s)) = ˆb ◦ ψ ◦ b∗ (s) ◦ t−bx = φ ◦ ˆb ◦ b∗ (s) ◦ t−x = φ ◦ s ◦ b ◦ t−x = φ ◦ s ◦ t−bx ◦ b = φ(s) ◦ b. Definition 7.2. Set ˆ 0 (L) ' lim H 0 (a∗ L), ιa : H 0 (a∗ L) → H ˆ 0 (L), H −→ a∈R
ˆ with ιa denoting the canonical inclusion. For any α = (x, (φa )a∈Jx ) ∈ G(L), define ˆ 0 (L) → H ˆ 0 (L), Uα : H
Uα |H 0 (a∗ L) = ρ˜a (φa ), ∀a ∈ Jx .
Such a Uα is well defined by Lemma 7.1. Let L be very ample, choose a section s ∈ H 0 (L) and assume anything defined over H. Proposition 7.3. The map U : α → Uα is a morphism, in particular we have: Uτ (x) ◦ Uτ (y) (s) = e( x2 , y)Uτ (x+y) (s).
K-LATTICES AND THE ADELIC HEISENBERG GROUP FOR CM CURVES
15
Proof. By Theorem 5.9 ∗ ˆ G(L) |p−1 ' p−1 ∗ a G(a L) a (K(a L)) ∗ so U (p−1 a (K(a L))) |H 0 (a∗ L) factorizes through ρa :
U
∗ p−1 a (K(a L))
ˆ 0 (L)) GL(H
p
G(a∗ L)
ρa
GL(H 0 (a∗ L))
ˆ ˆ 0 (L) we can find For any pair α = (xa , φa ), α0 = (x0a , φ0a ) in G(L) and any s ∈ H 0 ∗ a ∈ R s.t. a ∈ Jx ∩ Jx0 and s ∈ ιa (H (a L)) so the fact that U is a morphism follows from the asserted factorization of U through ρa . Finally, Uτ (x) ◦ Uτ (y) (s) = e( x2 , y)Uτ (x+y) (s) follows from Proposition 5.12.
Recall [14], Definition 5.5: ˆ 0 (L) and assume l ∈ L(0)∗ , Definition 7.4. Fix x ∈ V (E), s ∈ ιb (H 0 (b∗ L)) ⊂ H also defined in H. We define the adelic theta function associated to s: θs : V (E) → H, in such a way that θs (x) = l φ−1 a if τ (x) = x, (φa )a∈Jx .
a ∗ s(xa ) , a ∈ Jx ∩ (b), b
Such a theta fuction is well defined by the following: a ∗ −1 c ∗ Lemma 7.5. If both a and c belong to Jx ∩(b) then φ−1 a (( b ) s(xa )) = φc (( b ) s(xc )).
Proof. Of course we may assume c | a so that a = kc. In order to prove the lemma it suffice to show −1 c ∗ a ∗ ˆ kˆ ◦ φa (φ−1 a (( b ) s(xa ))) = k ◦ φa (φc (( b ) s(xc ))).
But we have: a ∗ c ∗ c ∗ ˆ kc ∗ kˆ ◦ φa (φ−1 a (( b ) s(xa ))) = k(( b ) s(xa )) = ( b ) s(kxa ) = ( b ) s(xc )
and c ∗ ˆ −1 c ∗ kˆ ◦ φa (φ−1 c (( b ) s(xc ))) = φc ◦ k(φc (( b ) s(xc ))) c ∗ c ∗ = φc (φ−1 c (( b ) s(xc ))) = ( b ) s(xc ).
We recall some properties of adelic theta functions (see [14], Chap. 5): Proposition 7.6. (1)
θs (x) = l(Uτ (−x) s)
(2)
θUτ (y) s (x) = e(y, x2 )θs (x − y)
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F. D’ANDREA AND D. FRANCO
Proof. (1) Assume that τ (x) = (x, (φa )a∈Jx ). By Proposition 7.3, we have Uτ (x) ◦ Uτ (−x) = e( x2 , −x)Uτ (0) = id −1 so Uτ−1 (x) = Uτ (−x) with τ (−x) = (−x, (φa )a∈Jx ). In order to ease notations, if a 0 ∗ 0 ∗ ˆ (L) we set sa = ( ) s, if a ∈ (b), so we have: s ∈ ιb (H (b L)) ⊂ H b
Uτ (−x) s =
φ−1 a sa (xa ),
and θs (x) = l(φ−1 a sa (xa )) = l(Uτ (−x) s).
(2) It follows combining (1) with Proposition 7.3: θUτ (y) s (x) = l(Uτ (−x)◦τ (y) s) = e(y, x2 )l(Uτ (y−x) s) = e(y, x2 )θs (x − y).
Theorem 7.7. Keep notations as in Theorems 6.1 and 6.2 and consider embeddings E(C) → P2 , E 0 (C) → P2 commuting with the action of σ. Fix sections s ∈ H 0 (E, L) and s0 ∈ H 0 (E 0 , L0 ) corresponding to the same line in (P2 (K))∗ . Then we have: σ(θs (x)) = θs0 (σ(x)). ˆ 0 ), ∀x ∈ V (E) so τ 0 (σ(x)) = Proof. By Theorem 6.2, τ 0 (σ(x)) = σ(τ (x)) ∈ G(L σ (σ(x), (φa )a∈Jx ). Then we have: θs0 (σ(x)) = l((φσa )−1 (s0a (σ(xa )))) = l((φσa )−1 (σ(sa (xa )))) by definition of s and s0 , l((φσa )−1 (σ(s(xa )))) = l(σ((φa )−1 (s(xa )))) by definition of σ : G → G, and finally l(σ((φa )−1 (s(xa )))) = σ(l((φa )−1 (s(xa )))) = σ(θs (x)) since l is defined over K.
We have furthermore the important Corollary (see [14], Proposition 5.6): Corollary 7.8. Assume l and s defined over H. For any σ ∈ Gal(H, H) we have: σ(θs (x)) = θs (σ(x)). Definition 7.9. (Compare with [16], Theorem 8.2) Fix an authomorphism of the complex numbers σ, and assume σ |Kab = [t, K], σ |Qab = [r, Q] via Artin maps: [ · , K] : A∗K → Gal(Kab , K),
[ · , Q] : A∗Q → Gal(Qab , Q).
We define: 1
χσ = V (E) × V (E) → Q ,
χσ (x, y) =
r−1 e(x, y) . e(s−1 x, s−1 y)
We state our main result concerning the behaviour of adelic theta functions under automorphisms: Theorem 7.10. Let σ ∈ Aut(C) and let t and r be ideles of K and Q corresponding to σ via Artin maps. With notations as above we have: σ(θUτ (y) s (x)) = χσ (y, x2 )θUτ (t−1 y) s0 (t−1 x).
K-LATTICES AND THE ADELIC HEISENBERG GROUP FOR CM CURVES
17
Proof. By Proposition 7.6, θUτ (y) s (x) = e(y, x2 )θs (x − y), so Theorem 7.7 implies: σ(θUτ (y) s (x)) = σ(e(y, x2 ))θs (x − y) = r−1 e(y, x2 )θs0 (σ(x − y)) = r−1 e(y, x2 )θs0 (t−1 (x − y)), and we conclude by applying Proposition 7.6 once again.
Corollary 7.11. Assume l and s defined over H. For any σ ∈ Gal(H, H) we have: σ(θUτ (y) s (x)) = χσ (y, x2 )θUτ (t−1 y) s (t−1 x), where t and r be ideles of K and Q corresponding to σ via Artin map. Finally we may now apply our results to the set of arithmetic 1-dimensional K lattices modulo commensurability. For example we find: Theorem 7.12. Keep notations as above and denote by F the set of arithmetic 1-dimensional K lattices modulo commensurability. Then, there exist adelic theta functions on F θs : F → H, supporting an action of the adelic Heisenberg group and such that: σ(θUτ (y) s (x)) = χσ (y, x2 )θUτ (t−1 y) s (t−1 x), for any σ ∈ Gal(H, H). Proof. By Corollary 4.9 (3), the set of arithmetic 1-dimensional K lattices modulo C commensurability maps to R . The conclusion then follows from Corollaries 7.8 and 7.11. References [1] Ch. Birkenhake, H. Lange: Complex Abelian Varieties. Grundlehren der Mathematischen Wissenschaften 302. Berlin: Springer (2004). [2] J.B. Bost, A. Connes: Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory. Sel. Math., New Ser. 1, No. 3, 411-457 (1995). [3] J.W.S. Cassels: Local fields. London Mathematical Society Student Texts, 3. Cambridge etc.: Cambridge University Press (1986). [4] P.B. Cohen: A C ∗ -dynamical system with Dedekind zeta partition function and spontaneous symmetry breaking. J. Thor. Nombres Bordx. 11, No.1, 15-30 (1999). [5] A. Connes, M. Marcolli: Noncommutative Geometry, Quantum Fields and Motives. Colloquium Publications, 55. American Mathematical Society (2005). [6] A. Connes, M. Marcolli, N. Ramachandran: KMS states and complex multiplication. Sel. Math., New Ser. 11, No. 3-4, 325-347 (2005). [7] A. Connes, M. Marcolli, N. Ramachandran: KMS states and complex multiplication. II. Bratteli, Ola (ed.) et al., Operator algebras. The Abel symposium 2004. Proceedings of the first Abel symposium, Oslo, Norway, September 3–5, 2004. Berlin: Springer. Abel Symposia 1, 15-59 (2006). [8] D. Harari, E. Leichtnam: Extension du phnomne de brisure spontane de symtrie de BostConnes au cas des corps globaux quelconques. Sel. Math., New Ser. 3, No.2, 205-243 (1997). [9] M. Laca, M. van Frankenhuijsen: Phase transitions on Hecke C ∗ -algebras and class-field theory over Q. J. Reine Angew. Math. 595, 25-53 (2006). [10] S. Lang: Algebraic Number Theory. Graduate Texts in Mathematics. 110. New York: Springer-Verlag (1994). [11] S. Lang: Elliptic Functions. Graduate Texts in Mathematics, 112. New York etc.: SpringerVerlag (1987).
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[12] D. A. Marcus: Number Fields. Univeritext. New York etc.: Springer-Verlag (Second Printing, 1987). [13] D. Mumford: On the Equations Defining Abelian Varieties, I-III. Invent. Math. 1, 287-354 (1966); ibid. 3, 75-135, 215-244 (1967). [14] D. Mumford: Tata lectures on theta III. In collaboration with Madhav Nori and Peter Norman. Reprint of the 1991 edition. Modern Birkhuser Classics. Basel: Birkhuser (2007). [15] L. Ribes, P. Zalesskii: Profinite groups. 2nd ed. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 40. Berlin: Springer (2010). [16] J. H. Silverman: Advanced Topics in the Arithmetic of Elliptic curves. Graduate Texts in Mathematics. 151. New York, NY: Springer-Verlag (1994). [17] G. Shimura: Introduction to Arithmetic theory of Automorphic Functions. Publications of the Mathematical Society of Japan. 11. Princeton, NJ: Iwanami Shoten, Publishers and Princeton University Press (1971). ` di Napoli “Federico II”, Dipartimento di Matematica e Ap(F. D’Andrea) Universita plicazioni “R. Caccioppoli” and I.N.F.N. Sezione di Napoli, Complesso Monte S. Angelo, Via Cinthia, 80126 Napoli. E-mail address:
[email protected] ` di Napoli “Federico II”, Dipartimento di Matematica e Appli(D. Franco) Universita cazioni “R. Caccioppoli”, Complesso Monte S. Angelo, Via Cinthia, 80126 Napoli. E-mail address:
[email protected]
