Galois action on the L-isogeny classes of abelian surfaces with quaternionic multiplication

Q

OF ABELIAN L-SURFACES WITH QUATERNIONIC MULTIPLICATION

SANTIAGO MOLINA

Abstract. We construct a projective Galois representations attached to an abelianL-surface with quaternionic multiplication, describing the Galois action on its Tate module. We prove that such representation characterizes the Galois action on the isogeny class of the abelianL-surface, seen as a set of points of certain Shimura curves.

1. Introduction

Let L be a number field. An abelian variety A/L¯ is called aabelian L-variety if for each σ ∈ Gal( ¯L/L)there exists an isogeny µ_σ : A^σ→A such thatψ◦µ_σ = µ_σ ◦ψ^σ for all ψ ∈ End(A). The current interest on abelian L-varieties began, withL=Q, when K. Ribet observed that absolutely simple factors of the modular Jacobians J1(N) are in fact abelian Q-varieties [2]. Actually, after the proof of Serre’s conjecture on representations ofGal( ¯Q/Q)[3, 3.2.4?], every so-calledbuilding blocks (namely thoseQ-varieties whose endomorphism algebra is a central division algebra over a totally real number field F with Schur index t = 1 or t = 2 and t[F : Q] = dimA), is an absolutely simple factors up to isogeny of a modular Jacobian J1(N).

In the dimension one case, we can think elliptic L-curves as generalizations of elliptic curves defined overL. Indeed, given an ellipticL-curveE(without Complex Multiplication) we can define a projective representation

(1.1) ρ^{`</sup><sub>E</sub> : Gal( ¯L/L)−→GL2(Q<sup>`})/Q^×`,

generalizing the classical representation on the Tate module of an elliptic curve over L.

In the dimension two case, we have a similar situation with the so-called fake elliptic curves or abelian surfaces with quaternionic multiplication (QM), namely, pairs (A, ı) where A is an abelian surface and ı is an embedding of a quaternion orderOinto its endomorphism ring. If we setA=E×EwithO= M2(Z)and the obvious embedding intoEnd(E×E), we recover the classical dimension one setting.

In this paper we shall construct a mapρ(A,ı), attached to an abelianL-surface with QM (A, ı), of the form

ρ_(A,ı): Gal( ¯L/L)−→(O ⊗Af)^×/(End(A, ı)⊗Q)^×

where Af is the ring of finite adeles and End(A, ı) is the set of endomorphisms commuting every element of the image of ı. The `-adic component ofρ<sub>(A,ı)</sub> will coincide with the projective representation of (1.1) in the classical scenario. As well as in this classical setting, the map ρ<sub>(A,ı)</sub> will depend on the choice of a basis of all Tate modulesT<sub>`</sub>A. We will see that this choice is equivalent to the choice of an isomorphism ofO-modulesϕ:Ator →B/O(Lemma 2.2), whereAtor is the set of torsion points of A andB =O ⊗Qis the corresponding quaternion algebra over Q. Hence, once the choice of ϕis provided, we will denote the corresponding map as ρ_(A,ı,ϕ).

1

It is easy to deduce that, given an abelian L-surface with QM(A, ı), any pair (A⁰, ı⁰) ¯L-isogenious to (A, ı) is also an abelian L-surface with QM. This implies that the L-isogeny class of¯ (A, ı)has a well defined action ofGal( ¯L/L). The main goal of this paper is to describe such Galois action by means of the mapρ_(A,ı). In order to do that, we will consider theL-isogeny class of¯ (A, ı)as a set ofL-rational¯ points of the certain Shimura curveX_Γ. By means of the moduli interpretation of X_Γ, the Galois action on the isogeny class will be translated into a Galois action on this set of infinitely many points. We will characterize this set of points as a double coset space (Proposition 3.3) and finally, in Theorem 4.2, we give a description of the Galois action by means of the mapρ_(A,ı).

The article is structured as follows: In §2 we introduce abelianL-surfaces with QM and the map ρ_(A,ı,ϕ), we give some of its properties distinguishing between the Complex Multiplication (CM) case and the non-CM case. In §3 we introduce Shimura curvesXΓ and their moduli interpretation, moreover, we characterize the isogeny class of an abelian surface with QM as a double coset space with a concrete embedding intoXΓ. In §4 we present our main result describing the Galois action on the isogeny class of an abelianL-surface with QM (A, ı) by means of the map ρ(A,ı,ϕ). In §5 we discuss about distinct moduli problems that the Shimura curve X_Γ solves and the reinterpretation of our main result in these new terms. Finally, in §6 we give a complete description of the CM case characterizing the mapρ_(A,ı,ϕ) via Class Field Theory.

1.1. Notation. Let ˆ

Zdenote the completion ofZ, hence ˆ Z= lim

←−(Z/NZ). LetA^f denote the ring of finite adeles, namelyAf = ˆZ⊗Q. Note thatQ/Z= lim

−→Z/NZ, therefore

End(Q/Z) = Hom(lim

−→(Z/NZ),Q/Z) = lim

←−Hom(Z/NZ,Q/Z)

= lim

←−Hom(Z/NZ,Z/NZ) = lim

←−(Z/NZ) = ˆZ.

LetBbe an indefinite quaternion algebra overQof discriminantD, and letObe an Eichler order inB. LetGbe the group scheme overZsuch thatG(R) = (O^opp⊗R)^× for all ringsR, whereO^oppis the opposite algebra toO. Note that the groupG(Af) does not depend on the Eichler order O chosen since it is maximal locally for all but finitely many places.

WriteOˆ =O ⊗Zˆ, then me have the isomorphism Oˆ = lim

←−(O/NO). Moreover lim−→(O/NO) = B/O as O-modules. By the same argument as above, this implies thatEnd_O(B/O) = ˆO^opp. Hence we can identifyG(Af) = (End_O(B/O)⊗Q)^×.

Throughout this paper, we will denoteEnd⁰:= End⊗Q.

2. AbelianL-surfaces with Quaternionic Multiplication

Anabelian surface with QM byO is a pair (A, ı)where A is an abelian surface andıis an embeddingO,→End(A), optimal in sense thatı(B)∩End(A) =ı(O).

If the orderO is clear by the context, we will call them justQM-abelian surfaces.

Let us consider the subring

End(A, ı) ={λ∈End(A) : λ◦ı(o) =ı(o)◦λ, for allo∈ O}.

IfA is defined over C, End(A, ı)⁰ can be eitherQ or an imaginary quadratic field K, in this last situation we say that(A, ı)hascomplex multiplication (CM).

Definition 2.1. LetM/L/Qbe number fields. AabelianL-surface with QM byO is an abelian surface with QM byO(A, ı)overM such that, for allσ∈Gal(M/L), there exist an isogenyµσ:A^σ →A, such thatµσ◦ı(o)^σ=ı(o)◦µσ for allo∈ O.

Given an abelianL-surface with QM we shall construct a map ρ_(A,ı,ϕ): Gal( ¯L/L)−→G(Af)/End⁰(A, ı)^×

that describes the Galois action on the Tate module. In order to do this, we will fix an isomorphism ϕ : A_tor → B/O. The following result shows that to choose such an isomorphismϕis equivalent to choose a basis{ϕ₁, ϕ₂, ϕ₃, ϕ₄}of the Tate moduleTˆ(A) = Hom(Ator,Q/Z).

Lemma 2.2. Given a basis{ϕ_i}_i=1···4 ofTˆ(A), there exists aZ-basis{e_i}_i=1···4 of O such that

ϕ:A_tor−→B/O; P 7−→ X

i=1···4

ϕ_i(P)e_i

is a O-module isomorphism. Analogously, given a O-module isomorphism ϕ : Ator → B/O, any Z-basis {ei}_i=1···4 of O provides a basis {ϕi}_i=1···4 of Tˆ(A) satisfying ϕ(P) =P4

i=1ϕ_i(P)e_i.

Proof. Since{ϕi}i=1···4is a basis, the mapAtor →(Q/Z)⁴is an isomorphism. Then there exists a unique sequence {Pn}n∈N ⊂Ator such thatϕi(Pn) = _n¹δ₁ⁱ. SinceA has QM by O, we know that A[n] ' O/nO. Thus there exists ej ∈ Osuch that ϕi(ı(ej)Pn) = _n¹δ_jⁱ (in particulare1 = 1). We claim that{ei}_i=1···4 form a Z-basis forO. Indeed, for anyα∈ O, there existsni∈Zsuch thatϕi(ı(α)Pn) =ⁿ_nⁱ, thus

ϕ_i(ı(α)P_n)−ϕ_i



ı



 X

j=1···4

n_je_j



P_n



=n_i

n − X

j=1···4

n_j nδ^j_i = 0.

Since {ϕi}i=1···4 form a basis and Pn generates A[n] as O-module, we conclude α=P

j=1···4njej and{ei}i−1···4 form aZ-basis forO.

Finally, we consider the well defined morphismϕand letα=P

j=1···4njej ∈ O, then

ϕ(ı(α)Pn) = X

i,j=1···4

njϕi(ı(ej)Pn)ei= X

i,j=1···4

nj

1

nδ_i^jei= 1 n

X

i=1···4

niei=αϕ(Pn).

SincePn generatesA[n]as aO-module, we conclude that ϕis aO-module isomor- phism.

Analogously, given aO-module isomorphismϕ:Ator→B/Oand given aZ-basis {ei}_i=1···4 ofO, we define the morphismϕi(P) =xi, where ϕ(P) =P

i=1···4xiei. It is clear that {ϕi}_i=1···4 provides aZ-basis ofHom(Ator,Q/Z).

Since A/M is an abelianL-surface with QM, we can fix isogeniesµσ:A^σ →A, for allσ∈Gal( ¯L/L). We denoteρ¯_(A,ı,ϕ)(σ)the element in End(B/O)satisfying

ϕ(µ_σ(P^σ)) =ϕ(P) ¯ρ_(A,ı,ϕ)(σ), for allP ∈A_tor. We compute that, for all α∈ O,

(αϕ(P)) ¯ρ_(A,ı,ϕ)(σ) = ϕ(µσ((ı(α)P)^σ)) =ϕ(µσ(ı(α)^σ(P^σ))) =ϕ(ı(α)µσ(P^σ))

= αϕ(µσ(P^σ)) =α ϕ(P) ¯ρ_(A,ı,ϕ)(σ) .

Thusρ¯_(A,ı,ϕ)(σ)∈End_O(B/O). Sinceµ_σhas finite kernel, we deduce thatρ¯_(A,ı,ϕ)(σ)∈ End⁰_O(B/O)^×. Once we identify End⁰_O(B/O)^× withG(Af), we have a map

¯

ρ_(A,ı,ϕ): Gal( ¯L/L)−→G(Af).

This map may depend on the choice of the isogenies µσ, nevertheless we can con- sider the quotientG(Af)/End⁰(A, ı)^×, whereEnd⁰(A, ı)^×is embedded inG(Af)by means of the natural embedding

ϕ^∗: End⁰(A, ı)^×,→G(Af) = End⁰_O(B/O)^×; ϕ^∗(λ) =λ^∗=ϕ◦λ◦ϕ⁻¹. Hence the composition with the quotient map, gives rise to a map of the form:

ρ(A,ı,ϕ): Gal( ¯L/L)−→G(Af)/End⁰(A, ı)^×.

Lemma 2.3. The mapρ(A,ı,ϕ) is independent on the choice of the isogeniesµσ.

Proof. Let µ⁰_σ : A^σ → A be another isogeny. Then λ_σ := _deg(µ¹

σ)µ⁰_σ ◦µ^∨_σ ∈ End⁰(A, ı)^×. We denote by ρ⁰_(A,ı,ϕ) the element in G(A^f) obtained using µ⁰_σ in- stead ofµ_σ. Thus we have:

ϕ(P) ¯ρ⁰_(A,ı,ϕ)(σ) = ϕ(µ⁰_σ(P^σ)) =ϕ(λσ(µσ(P^σ))) =ϕ(µσ(P^σ))λ^∗_σ

= ϕ(P) ¯ρ_(A,ı,ϕ)(σ)λ^∗_σ. Thusρ¯⁰_(A,ı,ϕ)(σ) = ¯ρ_(A,ı,ϕ)(σ)λ^∗_σ and

¯

ρ⁰_(A,ı,ϕ)(σ)End⁰(A, ı)^×= ¯ρ_(A,ı,ϕ)(σ)End⁰(A, ı)^×,

which proves our assertion.

Note that G(Af)/End⁰(A, ı)^× is a group in the non-CM case. Nevertheless, in the CM case, End(A, ı)⁰ = K an imaginary quadratic field, hence K^× it is not normal inB(Af)^×.

The embeddingEnd(A, ı),→ O^oppgives rise to an embedding of groups,End(A, ı),→ G(R), for all ringsR. Let us denote byN_A/Z be the normalizer group scheme of End(A, ı) inG, namely, the group scheme over Zsuch thatN_A(R) is the normal- izer of End(A, ı) in G(R). Note thatNA = G, in the non-CM case. Moreover, if (A, ı)has CM by the imaginary quadratic fieldK, thenNA(Q) =K^×∪jK^×, with j² ∈Q^× and jk= ¯kj for all k ∈K^×. In any case NA(Af)/End⁰(A, ı)^× is now a group.

Lemma 2.4. The map ρ(A,ı,ϕ) factors through ρ_(A,ı,ϕ): Gal( ¯L/L)^ρ

N (A,ı,ϕ)

−→ N_A(Af)/End⁰(A, ı)^× ,→G(Af)/End⁰(A, ı)^× Moreover, the mapρ^N_(A,ı,ϕ) is a group homomorphism.

Proof. On the one side, for allσ∈Gal( ¯L/L)andλ∈End(A, ı)we have

¯

ρ_(A,ı,ϕ)(σ)λ^∗ρ¯_(A,ı,ϕ)(σ)⁻¹∈End(A, ı)⁰. Indeed,

(degµσ)ϕ(P) ¯ρ_(A,ı,ϕ)(σ)λ^∗ρ¯_(A,ı,ϕ)(σ)⁻¹ = (degµσ)ϕ(λ(µσ(P^σ))) ¯ρ_(A,ı,ϕ)(σ)⁻¹

= ϕ(µ^∨_σ(λ(µσ(P^σ)))^σ−1)

= ϕ((µ^∨_σλµ_σ)^σ−1(P))

= ϕ(P)

(µ^∨_σλµσ)^σ−1∗

, where clearly(µ^∨_σλµ_σ)^σ−1∈End⁰(A, ı). Thereforeρ¯_(A,ı,ϕ)(σ)∈N_A(Af).

On the other side, one checks that

¯

ρ(A,ı,ϕ)(στ)⁻¹ρ¯(A,ı,ϕ)(σ) ¯ρ(A,ı,ϕ)(τ) acts onT Aˆ ⊗Q:= (Q0

pTpA)⊗Qin the same way as does

c_(A,ı)(σ, τ) = (1/deg(µστ))µσµ^σ_τµ_στ^∨ ∈(End(A, ı)⊗_ZQ)^×= End⁰(A, ı)^×. In particular, the quotientρ(A,ı,ϕ)(σ)is a group homomorphism.

Remark 2.5. Assume that the discriminant D = 1, thus the quaternion algebra B= M2(Q). An abelian surface with QM byO= M2(Z)is the productA=E×E, where E is an elliptic curve. In the particular case that E is defined over L(thus clearly A = E×E is an abelian L-surface with QM), the representation ρ_(A,ı,ϕ) is just the quotient moduloEnd⁰(A, ı)^×= End⁰(E)^× of the classical action on the Tate module

ρ_E: Gal( ¯L/L)−→GL₂(ˆZ) =Y

`

GL₂(Z`),→GL₂(Af).

3. Shimura curves and isogeny classes

Assume that O0 is a maximal order inB, letΓ be an open subgroup ofOˆ^×₀ = G(ˆZ). We say that two O0-module isomorphisms ϕ, ϕ⁰ : Ator

→' B/O0 are Γ- equivalent if there exists a γ ∈ Γ such that ϕ⁰ = ϕγ. The Shimura curve XΓ, is the compactification of the coarse moduli space of triples (A, ı,ϕ), where¯ (A, ı) are abelian surfaces with QM by O0 and ϕ¯ is a Γ-equivalence class of O0-module isomorphisms ϕ : A_tor →^' B/O₀. The curve X_Γ is defined over some number field L_Γ. If k is a field of characteristic zero, given P ∈ X_Γ(¯k) corresponding to the isomorphism class of a triple (A, ı,ϕ)/¯ ¯k, its Galois conjugate P^σ ∈X_Γ(¯k), for σ∈Gal(¯k/k), corresponds to the isomorphism class of(A^σ, ı^σ,ϕ¯^σ), where

ϕ^σ:A^σ_tor−→^' B/O0; ϕ^σ(Q^σ) =ϕ(Q).

Thus, ak-rational pointP onX_Γ corresponds to the isomorphism class of a triple (A, ı,ϕ)/¯ ¯kwhich is isomorphic to all itsGal(¯k/k)-conjugates.

The complex points of the Shimura curve are in correspondence with the double coset space

XΓ(C) = (Γ_∞Γ\G(A))/G(Q)∪ {cusps}, Γ_∞=

a b

−b a

∈GL2(R)

. The triple (A_g, ı_g,ϕ¯_g) over C corresponding to g = (g_∞, g_f) ∈ G(A) is A_g :=

(B ⊗R)_g∞/I_gf, where I_gf = ˆO₀g_f ∩B and (B ⊗R)_g∞ = M₂(R) with complex structurehg_∞

hg∞ :C→M2(R); i7→g_∞⁻¹

1

−1

g_∞;

the embeddingıg:O0→End(Ag), is given byıg(α)(b⊗z) =αb⊗z; andϕ¯g is the Γ-equivalence class ofϕg : (Ag)tor =B/Ig_f →B/O0,ϕg(b) =bg_f⁻¹. We compute that

End⁰(Ag, ıg)^× = {γ∈AutB(B⊗R) : γIg_f ⊗Q=Ig_f ⊗Qandγhg∞=hg∞γ}

= {γ∈G(R) : γB=B andγhg∞γ⁻¹=hg∞}

= {γ∈G(Q) : γhg_∞γ⁻¹=hg_∞}

= {γ∈G(Q) : g_∞γg_∞⁻¹∈Γ_∞}.

Remark 3.1. In most of the literature, objects classified by the Shimura curveXΓ

are triples(A, ı,ψ), where¯ (A, ı)is an abelian surface with QM byO0as above andψ¯ is aΓ-equivalence class ofO₀-module isomorphismsψ: ˆT(A) = Hom(A_tor,Q/Z)→^' Oˆ0. It is clear that this interpretation is equivalent to ours, since for anyϕ:Ator

→'

B/O0 we have the corresponding isomorphism

ψ: ˆT(A) = Hom(A_tor,Q/Z)−→^' Hom(B/O0,Q/Z)'Oˆ0.

Remark 3.2. In the particular case thatΓ = ΓN = ker(G(ˆZ)→ G(Z/NZ)), to give a Γ-equivalence class of isomorphisms ϕ: Ator → B/O0 is equivalent to give an isomorphism ϕN : A[N] → O0/NO0, namely, a level-N-structure. This is the classical Shimura curve situation.

We say that two triples (A, ı,ϕ)¯ and (A⁰, ı⁰,ϕ¯⁰) are isogenious if there exist an isogeny φ: A⁰ → A, satisfying φ◦ı⁰(α) = ı(α)◦φ, for all α∈ O. We denote by φ: (A⁰, ı⁰)→(A, ı)the isogeny with the corresponding compatibility with respect to ıand ı⁰.

Let P ∈ XΓ(C) be a point corresponding to (A, ı,ϕ). Let us denote by¯ [P]

the C-isogeny class of (A, ı,ϕ)/¯ C in XΓ, namely, the set of points Q ∈ XΓ(C) parametrizing triples (A⁰, ı⁰,ϕ¯⁰)/Cwhere(A⁰, ı⁰)is isogenious to(A, ı).

Proposition 3.3. Let P = [g] = [g_∞,1]∈(Γ_∞Γ\G(A))/G(Q)⊆X_Γ(C). Then we have the following bijection

ψ_g∞ : Γ\G(Af)/End⁰(A_g, ı_g)^× −→^' [P]; g_f 7−→[g_∞, g_f].

Proof. The non-CM case is described in [1, Lemma 1], we give here a proof that works in any case. Recall that (Ag∞, ıg∞,ϕ¯g∞) is the triple corresponding to P. For anygf∈G(Af), there existsn∈Zsuch thatIg_fn⊆ O0, therefore we have the isogeny

A_g∞gf = (B⊗R)_g∞/I_gf −→(B⊗R)_g∞/O₀=A_g∞, b7−→nb,

which is clearly compatible with the embeddingsıg∞ andıg∞g_f since the inclusion Ig_fn⊆ O0 is a monomorphism of O0-modules. This implies [g∞, gf]∈[P], for all gf ∈G(Af).

Conversely, any isogeny(A_g⁰_∞gf, ı_g∞⁰ _gf)→(A_g∞, ı_g∞)induces an equality of com- plex structures(B⊗R)_g⁰_∞= (B⊗R)_g∞, that implies thatg_∞⁰ = Γ_∞g_∞. Therefore the corresponding point[g⁰_∞, g_f]has a representant of the form[g_∞, g_f⁰]in the double coset space(Γ_∞Γ\G(A))/G(Q).

We conclude that we have a surjective map

Γ\G(Af)−→[P]; gf 7−→[g∞, gf],

and the result follows from the fact that[g_∞, gf] = [g_∞, g_f⁰]in (Γ_∞Γ\G(A))/G(Q) if and only if there existsβ ∈G(Q)such thatgf =g_f⁰β andg_∞β ∈Γ_∞g_∞, hence

β∈End⁰(A_g∞, ı_g∞)^×.

Remark 3.4. Let us consider the natural map

X_Γ⊇(Γ_∞Γ\G(A))/G(Q)−→^ψ Γ_∞\G(R)/G(Q), [g_∞, g_f]7−→[g_∞].

Then it is clear that, if P = [g_∞, gf], the isogeny class [P] is the fiber of ψ over [g∞].

4. Galois action on isogeny classes

Assume now that(A, ı) is an abelianL-surface with QM byO0, let (A, ı,ϕ)¯ be a triple corresponding to the point P ∈ XΓ( ¯Q). First we show that any (A⁰, ı⁰) isogenous to(A, ı)is an abelianL-surface with QM.

Lemma 4.1. Let(A, ı)/L¯be an abelianL-surface with QM and assume that(A⁰, ı⁰)/L¯ is isogenous to (A, ı). Then(A⁰, ı⁰)/L¯ is an abelian L-surface with QM.

Proof. Letσ∈Gal( ¯L/L). Since(A, ı)is aLabelian surface with QM, there exists an isogeny(A^σ, ı^σ)^µ→^σ(A, ı). Fix an isogeny(A⁰, ı⁰)→^φ (A, ı). Thus by conjugating φbyσand composing withφ^∨◦µ_σ, one obtains

((A⁰)^σ,(ı⁰)^σ) ^φ

σ

−→(A^σ, ı^σ)−→^µσ (A, ı) ^φ

∨

−→(A⁰, ı⁰).

Hence(A⁰, ı⁰)/L¯ is an abelianL-surface with QM.

Note that, sinceP and so(A, ı)are defined over ¯

Q, theC-isogeny class coincide with the ¯

Q-isogeny class [P]. Moreover, the above lemma implies that Gal( ¯L/L) acts on [P]. Indeed, ifQ∈[P] corresponds to(A⁰, ı⁰,ϕ¯⁰) andσ∈Gal( ¯L/L), then Q^σparametrizes((A⁰)^σ,(ı⁰)^σ,( ¯ϕ⁰)^σ). Since(A⁰, ı⁰)is an L-abelian surface with QM by the lemma, there existsµ⁰_σ : ((A⁰)^σ,(ı⁰)^σ)→(A⁰, ı⁰). This implies ((A⁰)^σ,(ı⁰)^σ) is isogenous to(A, ı), henceQ^σ∈[P]. The main theorem of this section relates this action with the map ρ_(A,ı,ϕ) introduced in §2 by means of the characterization of [P]given in Proposition 3.3.

Theorem 4.2. Assume that P = [g_∞,1] ∈ X_Γ corresponds to a triple (A, ı,ϕ),¯ where (A, ı)/Q¯ is an abelian L-surface with QM andϕ¯ is theΓ-equivalent class of the natural isomorphism

ϕ:A_tor= ((B⊗R)_g∞/O0)_tor−→B/O0,

Then the map ρ(A,ı,ϕ): Gal( ¯L/L)−→G(Af)/End⁰(A, ı)^× constructed by means of ϕ satisfies

ψ_g∞(g_f)^σ=ψ_g∞(g_fρ_(A,ı,ϕ)(σ))∈[P], for all g_g∈G(Af)andσ∈Gal( ¯L/L).

Remark 4.3. Note that, by Lemma 2.4, the image ρ_(A,ı,ϕ)(σ) lies in the com- mutator of End⁰(A, ı) in G(Af), thus the product gfρ(A,ı,ϕ)(σ) is well defined in Γ\G(Af)/End⁰(A, ı).

Proof. Recall that the abelian surface corresponding to ψg∞(gf) is given by the complex torusAg_f = (B⊗R)g∞/Ig_f, whereIg_f =B∩Oˆ0gf. Moreover, considering a representative ofΓgfEnd⁰(A, ı)^×such thatg_f⁻¹∈Oˆ0, theO0-stable isogeny between A andAg_f is given by

φg_f :A= (B⊗R)g_∞/O0−→(B⊗R)g_∞/Ig_f =Ag_f, b7−→b.

Also recall that a representative of ϕ¯g_f is given by

ϕg_f : (Ag_f)tor = ((B⊗R)g_∞/Ig_f)tor=B/Ig_f −→B/O0, b7−→bg_f⁻¹. Thus one checks thatϕ_gf ◦φ_gf =ϕg_f⁻¹:A_tor→B/O₀.

For anyσ∈Gal( ¯L/L), the pointψg∞(gf)^σcorresponds to the triple(A^σ_gf, ı^σ_gf,ϕ¯^σ_gf).

We have the following isogenies (A^σ_g

f, ı^σ_g

f)

φ^σ_gf

←−(A^σ, ı^σ)−→^µσ (A, ı)

φ_gf

−→(Ag_f, ıg_f),

thus(A^σ_gf, ı^σ_gf)and(A, ı)are linked by the isogenyφ^σ_gf◦µ^∨_σ. This implies that, for allP ∈Ator, we haveϕ(P)ψ⁻¹_g∞(ψg_∞(gf)^σ)⁻¹=ϕ^σ_gf(φ^σ_gf(µ^∨_σ(P))), hence

ϕ(µ_σ(P^σ))ψ_g⁻¹_∞(ψ_g∞(g_f)^σ)⁻¹ = deg(µ_σ)ϕ^σ_g

f(φ^σ_g

f(P^σ)) ϕ(P)ρ_(A,ı,ϕ)(σ)ψ_g⁻¹_∞(ψg∞(gf)^σ)⁻¹ = deg(µσ)ϕ^σ_gf(φg_f(P)^σ)

= deg(µσ)ϕg_f(φg_f(P))

= deg(µ_σ)ϕ(P)g_f⁻¹

We conclude gfρ(A,ı,ϕ)(σ) =ψ_g⁻¹_∞(ψg∞(gf)^σ), thusψg∞(gfρ(A,ı,ϕ)(σ)) =ψg∞(gf)^σ. 5. Change of moduli interpretation

In section §2, we defined an abelian L-surface (A, ı) with QM by any Eichler order O and defined the corresponding representationρ_(A,ı,ϕ) attached to a fixed O-module isomorphismϕ: Ator → B/O. Nevertheless, we used a maximal order O0 to define the Shimura curveXΓ and to describe its moduli interpretation as the space classifying triples (A0, ı0,ϕ¯0), where(A0, ı0) has QM byO0. In this section we shall change this moduli interpretation for some of this Shimura curves XΓ in order to classify abelian surfaces with QM by O.

Thus from now onOwill be an Eichler order inB of levelN andO0a maximal order such that O ⊆ O0. Fix the embedding λ:O,→ O0. Let Γ be now an open subgroup ofOˆ^×= (O ⊗ˆ

Z)^×. SinceOˆ^× is an open subset ofG(ˆZ) = ˆO₀^× by means of λ, the subgroup Γ is also an open subgroup of G(ˆZ). Thus we can consider the Shimura curve XΓ.

Proposition 5.1. We have an equivalence of moduli interpretations for the Shimura curveX_Γ. It either classifies:

(i) Triples (A₀, ı₀,ϕ¯₀), where (A₀, ı₀) is an abelian surface with QM by O0

andϕ¯₀ is aΓ-equivalence class ofO0-module isomorphismsϕ₀: (A₀)_tor→ B/O₀.

(ii) Triples(A, ı,ϕ), where¯ (A, ı)is an abelian surface with QM by O andϕ¯ is aΓ-equivalence class of O-module isomorphismsϕ:A_tor→B/O.

In order to prove this proposition we will need the following lemma. Note that the embeddingλ:O,→ O0gives rise to a morphismλ:B/O →B/O0.

Lemma 5.2. There exists a one-to-one correspondence between triples (A, ı, ϕ), where(A, ı)is an abelian surface with QM byOandϕis aO-module isomorphism ϕ:A_tor →B/O, and triples (A₀, ı₀, ϕ₀), where (A₀, ı₀)is an abelian surface with QM by O₀ and ϕ₀ is a O₀-module isomorphism ϕ₀ : (A₀)_tor → B/O₀. A triple (A, ı, ϕ)corresponds to (A₀, ı₀, ϕ₀)if there exists an isogenyφ:A→A₀, such that ϕ₀◦φ=λ◦ϕandφ◦ı(α) =ı₀(λ(α))◦φ, for allα∈ O.

Proof. Given(A, ı, ϕ), let us consider the subgroupC:=ϕ⁻¹(ker(B/O→^λ B/O0))⊂ A_tor. Therefore, we can construct the abelian surface A₀ = A/C and the corre- sponding isogeny φ: A →A₀. Since O ⊆ O₀, for allα∈ O, we have α(kerλ)⊆ kerλ, hence,ı(α)C ⊆C and the embeddingıgives rise to an embedding ı₀:O,→ End(A0). Moreover, theO-module isomorphimϕgives rise to aO-module isomor- phismϕ0 that fits into the following commutative diagram:

Ator

ϕ //

φ

B/O

λ

(A0)tor=Ator/C ^ϕ0 //B/O0

Henceϕ0◦φ=λ◦ϕ. Moreover, the fact that(A0)tor'B/O0asO-modules implies that ı0 can be extended to an embeddingı0 : O0 ,→ End(A0). Thus (A0, ı0)has QM byO0. We have constructed the triple(A0, ı0, ϕ0)corresponding to(A, ı, ϕ).

Finally, given(A0, ı0, ϕ0), let us considerC^∨:=ϕ⁻¹₀ (ker(B/O0 λ^∨

→B/O)), where λ^∨ : B/O0 →B/O is given byλ^∨(b) = [O0 : O]λ⁻¹(b). We define A :=A0/C^∨, thus the isogeny with kernelC^∨is the dual isogeny of someφ:A→A0that fits into the above commutative diagram for someO-module isomorphismϕ. We construct the triple(A, ı, ϕ)corresponding to (A₀, ı₀, ϕ₀)as in the previous situation.

Due to this previous lemma we can easily prove the above proposition:

Proof of Proposition 5.1. We know that the Shimura curveX_Γclassify triples(A₀, ı₀,ϕ¯₀) as in(i). By the above Lemma, given a representativeϕ0of theΓ-equivalence class

¯

ϕ0, there exists a triple (A, ı, ϕ), where ϕis a O-module isomorphism. It is clear that theΓ-equivalence classϕ¯0 corresponds to aΓ-equivalence classϕ.¯ Definition 5.3. A triple with QM by O is a triple (A, ı, ϕ), where (A, ı) is an abelian surface with QM byOandϕis aO-module isomorphismϕ:Ator→B/O.

A L-triple with QM byO is a triple (A, ı, ϕ)with QM byO such that(A, ı)is an abelianL-surface with QM.

We denote the one-to-one correspondence of Lemma 5.2 by

Λ^O_O⁰ :{Triples with QM byO} −→ {Triples with QM byO0}

Note that, given a triple(A, ı, ϕ)with QM byO, one can construct the projective representation

ρ_(A,ı,ϕ): Gal( ¯L/L)−→G(Af)/End⁰(A, ı)^×.

The following result relates the representations attached to triples associated by the correspondence Λ^O_O⁰.

Lemma 5.4. Let(A, ı, ϕ)be aL-triple with QM byO. Assume thatΛ^O_O⁰(A, ı, ϕ) = (A0, ı0, ϕ0), then we have that(A0, ı0, ϕ0) is aL-triple with QM byO0 and

ρ_(A,ı,ϕ)=ρ_(A0,ı0,ϕ0),

when we identify G(A^f) = End⁰_O(B/O)^×= End⁰_O0(B/O0)^× by means ofλ.

Proof. We know that there exists an isogeny φ:A→A0, such that ϕ0◦φ=λ◦ϕ and φ◦ı(α) = ı0(λ(α))◦φ, for all α ∈ O. Since (A, ı) is an abelian L-surface with QM, there exists isogenies µσ : A^σ → A, for all σ ∈ Gal( ¯L/L), such that µσ◦ı(o)^σ=ı(o)◦µσ for allo∈ O. The composition

µ⁰_σ:A^σ₀ ^(φ

σ)^∨

−→ A^σ−→^µσ A−→^φ A₀, satisfies

µ⁰_σ◦ı0(λ(o))^σ = φ◦µσ◦(φ^σ)^∨◦ı0(λ(o))^σ=φ◦µσ◦ı(o)^σ◦(φ^σ)^∨

= φ◦ı(o)◦µσ◦(φ^σ)^∨=ı0(λ(o))◦µ⁰_σ, for allo∈ O, thusµ⁰_σ◦ı0(α)^σ=ı0(α)◦µ⁰_σfor allα∈ O0. This implies that(A0, ı0)is an abelian L-surface with QM.

Moreover, for all σ∈Gal( ¯L/L)andP ∈(A0)tor, we have

ϕ0(µ⁰_σ(P^σ)) = ϕ0(φ◦µσ◦(φ^σ)^∨(P^σ)) =λ(ϕ(µσ(φ^∨(P)^σ)))

= λ(ϕ(φ^∨(P)))ρ_(A,ı,ϕ)(σ) = deg(φ)ϕ0(P)ρ_(A,ı,ϕ)(σ).

This impliesρ(A,ı,ϕ)(σ) =ρ(A₀,ı₀,ϕ₀)(σ).

Remark 5.5. As a consequence of this lemma, we obtain that Theorem 4.2 also applies if we change the maximal order O₀ by a not necessarily maximal Eichler orderO, considering the moduli interpretation(ii)of Proposition 5.1.

6. Complex Multiplication abelian K-surfaces with QM

In this section we shall deal with the Complex Multiplication (CM) case. Hence, only for this section, we assume that the abelian surface(A, ı)with QM byOhas also CM byK, so is to sayEnd⁰(A, ı) =Kan imaginary quadratic field. The following result describes the projective representationρ^N_(A,ı,ϕ)(and therefore ρ(A,ı,ϕ)) in the CM case:

Proposition 6.1. Assume that End(A, ı) = OK, an order in K. We have the following results

(i) Any abelian surface (A⁰, ı⁰) with QM by O and CM by K is isogenious to (A, ı).

(ii) We can chose a representative of the isomorphism class of (A, ı) defined overQ¯. Moreover, it is an abelianQ-surface with QM.

(iii) Given any isomorphism ϕ : Ator → B/O, the restriction to Gal( ¯K/K) of the corresponding projective representation ρ^N_(A,ı,ϕ) factors through the inverse of the Artin map Art : A^×K,f/K^× −→ Gal(K^ab/K), namely, we have the following commutative diagram

Gal( ¯Q/Q)

ρ^N_(A,ı,ϕ)

//NA(Af)/K^×

Gal( ¯K/K)?OO ////Gal(K^ab/K) ^Art

−1 //A^×K,f?/KOO ^×.

(iv) The representation ρ^N_(A,ı,ϕ) factors throughGal(K^ab/K)oGal(K/Q)send- ing the generator ofGal(K/Q) to the class jK^×, where j ∈N_A(Q)is any element satisfying j²∈Q^× andjk= ¯kj, for allk∈K^×.

Proof. We have just seen that abelian surfaces with QM byOoverCare classified by the non-cuspidal points ofX_Oˆ×. Assume that(A, ı)with CM byKcorresponds to [g_∞, gf] ∈ (Γ_∞Oˆ^×\G(A))/G(Q). Thus the complex structure on its tangent space is given by an embeddinghg∞ :C,→B^opp⊗R. On the other side, we have the natural embeddingψ(A,ı): End(A, ı),→EndO(H1(A,Z)) = EndO(Ig_f) =O^opp, giving rise toψ(A,ı):OK ,→ O^opp, satisfying

ψ_(A,ı)(K)∩ O^opp = End⁰(A, ı)∩End_O(t_A) = End(A, ı) =ψ_(A,ı)(O_K).

Since any pair ψ(A,ı)(k),k∈K, and hg∞(z), z∈Ccommute inEndO(H1(A,Z)⊗ R) =B^opp⊗R, we have thathg∞ is the extension of scalars ofψ(A,ı).

Given another pair(A⁰, ı⁰)with CM by K corresponding to[g_∞⁰ , g⁰_f], by Skolem- Noether, there existsg ∈G(Q), such thatψ(A⁰,ı⁰) =g⁻¹ψ(A,ı)g. This implies that gg⁰_∞ = g⁻¹hg∞g and g_∞⁰ =g∞g, hence [g_∞⁰ , g⁰_f] = [g∞, g_f⁰g⁻¹]. We conclude that (A⁰, ı⁰)is isogenious to(A, ı)by Proposition 3.3. This proves(i).

Letσ∈Aut(C) and assume that(A, ı)corresponds to the double coset [g∞,1].

The pair (A^σ, ı^σ) satisfiesEnd(A, ı) = End(A^σ, ı^σ), thus it correspond to a point [g_∞, g_f]∈X_Oˆ× by(i). Fix aO-module isomorphism ϕ:A_tor→B/O. The choice of a representativeg_f such thatI_gf ⊆ Ofixes aO-module isomorphismϕ⁰:A^σ_tor→ B/O and an isogenyµ_σ : (A^σ, ı^σ)→(A, ı)such that ϕ(µ_σ(Q)) =ϕ⁰(Q)g_f. Since A^σ_tor is isomorphic to A_tor by means of the O-module isomorphism P 7→ P^σ, we have thatϕ(µσ(P^σ))∈ϕ(P) ˆO^×gf. We compute that, for everyλ∈End(A, ı),

µ^∨_σ(λ(µ_σ(P^σ)))^σ−1 = (µ^∨_σ◦λ◦µ_σ)^σ−1(P), (µ^∨_σ◦λ◦µ_σ)^σ−1 ∈End(A, ı), Therefore the map ϕ(P) 7→ ϕ(µ_σ(P^σ)) is in the commutator of O_K in O. Thisˆ implies that we can choose a representative of [gf] ∈ Oˆ^×\G(Af) in the commu- tator NA(Af) of End(A, ı) in G(Af). Hence we can suppose that gf ∈ Oˆ^× ∩ N_A(Af)\NA(Af)/K^×. This double coset space is the semi-direct product ofZ/2Z andOˆ_K^×\A^×K,f/K^×= Cl(OK), which is of couse finite. Thus the set of isomorphism classes {(A^σ, ı^σ) :σ ∈ Aut(C)} is finite. This implies that (A, ı) admits a model overQ¯. By(ii)we deduce that(A, ı)is an abelian Q-surface with QM, this proves (ii).

Part(iii) follows directly from Theorem 4.2 and Shimura’s Reciprocity Law [4, Main Theorem I].

Finally, note first that the classjK^×is an element inN_A(Af)/K^×of order 2 and NA(Af)/K^×=A^×_K,f/K^×ohjK^×i. Letσc ∈Aut(C)denote complex multiplication automorphism. Since (A, ı) is defined over ¯

Q, the automorphism σ_c acts as an element inGal( ¯Q/Q)on ¯

Q-rational points of(A, ı). Recall thatA= (B⊗R)_g∞/O, where the complex structure on(B⊗R)_g∞ is given byh_g∞, the scalar extension of ψ_(A,ı):OK ,→ O^opp. Complex conjugation must be the unique automorphismγon (B⊗R)such thatγ◦hg∞(z) =hg∞(z^σc)◦γ, for allz∈C^×. Thereforeγcorresponds to conjugate by j sinceψ_(A,ı)(k)j⁻¹bj =j⁻¹ψ_(A,ı)(k^σc)bj, for allb ∈B, k ∈K^×. This implies thatA^σc= (B⊗R)g∞j/O= (B⊗R)g∞/Oj and we have the following diagram:

Ator P7→P^σc//

ϕ

A^σ_tor^{c µσc} //

Ator ϕ

B/O

b7→bj//B/Oj

b7→b //B/O

Thusϕ(µ_σc(P^σc)) =ϕ(P)j, which impliesρ_(A,ı)(σ_c) =jK^×. Sinceρ^N_(A,ı,ϕ)maps ex- haustivelyGal( ¯K/K)toA^×_K,f/K^×by(iii), andGal( ¯Q/Q)/Gal( ¯K/K)'Gal(K/Q), which is generated by the image ofσ_c, part(iv)follows.

References

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[3] Jean-Pierre Serre. Sur les représentations modulaires de degré2deGal(Q/Q).Duke Math. J., 54(1):179–230, 1987.

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Department of Mathematics, Universität Bielefeld, Bielefeld (Germany) E-mail address:santimolin@gmail.com