The field of moduli of quaternionic multiplication on abelian varieties

We consider principally polarized abelian varieties with quaternionic multiplication over number fields and we study the field of moduli of their endomorphisms in relation to the set of rational points on suitable Shimura varieties.


Introduction
LetQ be a fixed algebraic closure of the field Q of rational numbers and let (A, L)/Q be a polarized abelian variety. The field of moduli of (A, L)/Q is the minimal number field k A,L ⊂ Q such that (A, L) is isomorphic (over Q) to its Galois conjugate (A σ , L σ ), for all σ ∈ Gal(Q/k A,L ).
The field of moduli k A,L is an essential arithmetic invariant of the Q-isomorphism class of (A, L). It is contained in all possible fields of definition of (A, L) and, unless (A, L) admits a rational model over k A,L itself, there is not a unique minimal field of definition for (A, L). In this regard, we have the following theorem of Shimura.
Theorem 1.1 ( [19]). A generic principally polarized abelian variety of odd dimension admits a model over its field of moduli. For a generic principally polarized abelian variety of even dimension, the field of moduli is not a field of definition.
Let End(A) = End Q (A) denote the ring of endomorphisms of A. It is well known that End(A) = Z for a generic polarized abelian variety (A, L). However, due to Albert's classification of involuting division algebras ( [13]) and the work of Shimura ([18], there are other rings that can occur as the endomorphism ring of an abelian variety. Namely, if A/Q is simple, End(A) is an order in either a totally real number field F of degree [F : Q] | dim(A), a totally indefinite quaternion algebra B over a totally real number field F of degree 2[F : Q] | dim(A), a totally definite quaternion algebra B over a totally real number field F of degree 2[F : Q] | dim(A) or a division algebra over a CM-field. 1 Let us recall that a quaternion algebra B over a totally real field F is called totally indefinite if B ⊗ Q R ≃ M 2 (R)⊕...⊕M 2 (R) and totally definite if B ⊗ Q R ≃ H⊕...⊕H, where H = ( −1,−1 R ) denotes the skew-field of real Hamilton quaternions. Definition 1.2. Let (A, L)/Q be a polarized abelian variety and let S ⊆ End(A) be a subring of End(A). The field of moduli of S is the minimal number field k S ⊇ k A,L such that, for any σ ∈ Gal(Q/k S ), there is an isomorphism ϕ σ /Q : A → A σ , ϕ * σ (L σ ) = L, of polarized abelian varieties that induces commutative diagrams We remark that, as a consequence of the very basic definitions, the field of moduli of the multiplication-by-n endomorphisms on A is exactly k Z = k A,L . But in the case that End(A) Z, little is known on the chain of Galois extensions k End(A) ⊇ k S ⊇ k A,L .
The main aim of this article is to study the field of moduli of totally indefinite quaternionic multiplication on an abelian variety. In relation to Shimura's Theorem 1.1, we remark that the dimension of an abelian variety whose endomorphism ring contains a quaternion order is always even.
We state our main result in the next section. As we will show in Section 3, it is a consequence of the results obtained in [16], [17] on certain modular forgetful morphisms between certain Shimura varieties, Hilbert modular varieties and the moduli spaces of principally polarized abelian varieties.
In Section 4, we particularize our results to abelian surfaces. We use our results together with those of Mestre [11] and Jordan [10] to compare the field of moduli and field of definition of the quaternionic multiplication on an abelian surface.
In an appendix to this paper, we discuss a question on the arithmetic of quaternion algebras that naturally arises from our considerations and which is also related to recent work by Chinburg and Friedman [2], [3].
A cryptographical application of the results in the appendix has been derived in [8] by Galbraith and the author.

Main result
Let F be a totally real number field F of degree [F : Q] = n and let R F denote its ring of integers. We shall let F * + denote the subgroup of totally positive elements of F * . For any finite field extension L/F , let R L denote the ring of integers of L and let Ω odd (L) = {ξ ∈ R L , ξ f = 1, f odd } denote the set of primitive roots of unity of odd order in L. We let ω odd (L) = |Ω(L)|.
Let B be a totally indefinite quaternion algebra over F and let O be a maximal order in B. Proposition 2.2. [15] Let (A, L) be a principally polarized abelian variety with quaternionic multiplication by O over Q. Then the discriminant ideal disc(B) of B is principal and generated by a totally positive element D ∈ F * + . As in [16], [17] we say that a quaternion algebra B over F of totally positive supported at the prime ideals ℘ | D of F . Let C 2 denote the cyclic group of two elements. The main result of this article is the following.
As we state more precisely in Section 3, Theorem 2.3 admits several refinements.

Proof of Theorem 2.3: Shimura varieties and forgetful maps
Let B be a totally indefinite quaternion division algebra over a totally real number field F and assume that disc(B) = (D) for some D ∈ F * + . Let O be a maximal order in B and fix an arbitrary quaternion µ ∈ O satisfying µ 2 + D = 0. Its existence is guaranteed by Eichler's theory on optimal embeddings ( [21]) and it generates a We will refer to the pair (O, µ) as a principally polarized order.
Attached to (O, µ), we can consider a Shimura variety X O,µ /Q that solves the coarse moduli problem of classifying triplets (A, ι, L) over Q where: (i) (A, L) is a principally polarized abelian variety and (ii) ι : O ֒→ End(A) is a monomorphism of rings satisfying that ι(β) * = ι(µ −1β µ) for all β ∈ O and where * denotes the Rosati involution with respect to L. Attached to the maximal order O there is also the Atkin-Lehner group Let C 2 denote the cyclic group of two elements. The group W is isomorphic to ... ×C 2 , where 2r = ♯{℘ | disc(B)} is the number of ramifying prime ideals of B (cf. [21], [16]).
Let B * + be the group of invertible quaternions of totally positive reduced norm. The positive Atkin-Lehner group is denotes the group of units of O of reduced norm 1.
As it was shown in [16], the group W 1 is a subgroup of the automorphism group Aut Q (X O,µ ) of the Shimura variety X O,µ .
We have W 1 ≃ C s 2 for 2r ≤ s ≤ n + 2r − 1. The first inequality holds because there is a natural map W 1 ։ W which is an epimorphism of groups due to indefiniteness of B and the norm theorem for maximal orders (see [16]). The second inequality is a consequence of Dirichlet's unit theorem and it is actually an equality if the narrow class number of F is h + (F ) = 1, as is the case of F = Q.
We now introduce the notion of twists of a polarized order (O, µ).
For any subring S ⊂ O, we say that χ is a twist of (O, µ) in S if χ ∈ S.
We say that (O, µ) is twisting if it admits some twist in O and that a quaternion algebra B is twisting if it contains a twisting polarized maximal order. This agrees with our terminology in the preceding section. Let V 0 (S) denote the subgroup of W 1 generated by the twisting involutions of (O, µ) in S; we will simply write V 0 for V 0 (O).
Let us remark that, since B is totally indefinite, no χ ∈ B * + can be a twist of (O, µ) because a necessary condition for B ≃ ( −D,−n(χ) F ) is that n(χ) be totally negative. In fact, twisting involutions ω ∈ W 1 are always represented by twists χ ∈ B * − of totally negative reduced norm.
Note also that a necessary and sufficient condition for B to be twisting is that B ≃ ( −D,m F ) for some element m ∈ F * + supported at the prime ideals ℘ | D (that is, v ℘ (m) = 0 only if ℘ | D).
denote the finite group of roots of unity in R µ and Ω odd = {ξ ∈ R µ , ξ f = 1, f odd } the subgroup of primitive roots of unity of odd order. Their respective cardinalities will be denoted by ω = ω(R µ ) and ω odd = ω odd (R µ ).
The motivation for introducing the Shimura variety X O,µ and the above Atkin-Lehner groups in this note is that it gives a modular interpretation of the field of moduli k O of the quaternionic multiplication on A: k O = Q(P ) is the extension over Q generated by the coordinates of the point P = [A, ι, L] on Shimura's canonical model X O,µ /Q that represents the Q-isomorphism class of the triplet.
A similar construction holds for the totally real subalgebras of B. Indeed, let L ⊂ B be a totally real quadratic extension of F embedded in B. Then S = L ∩ O is an order of L over R F which is optimally embedded in O. Identifying S with a subring of the ring of endomorphisms of A, we again have that the field of moduli k S is the extension Q(P |S ) of Q generated by the coordinates of the point P |S = [A, ι |S , L] on the Hilbert-Blumenthal variety H S /Q that solves the coarse moduli problem of classifying abelian varieties of dimension 2n with multiplication by S.
Along the same lines, the field of moduli k R F of the central endomorphisms of A is the extension Q(P |R F ) of Q generated by the coordinates of the point P |R F = [A, ι |R F , L] on the Hilbert-Blumenthal variety H F /Q which solves the coarse moduli problem of classifying abelian varieties of dimension 2n with multiplication by R F .
The tool for studying the Galois extensions k O /k S /k R F is provided by the forgetful modular maps It was shown in [16] that the morphisms π F and π S have finite fibres. Furthermore, it was proved in [16] that: We say that a closed point [A, ι, L] in X O,µ or in any quotient of it is a Heegner point if End(A) ι(O). It was also shown in [16] that the morphisms b F and b S are biregular on X O,µ /W 0 and X O,µ /V 0 (S), respectively, outside a finite set of Heegner points.
It follows from these facts that the Galois group G = Gal(k O /k R F ) of the extension of fields of moduli k O /k R F is naturally embedded in W 0 : any σ ∈ G acts on a principally polarized abelian variety with quaternionic multiplication (A, ι : In what follows, we will describe the structure of the groups W 0 and V 0 (S) attached to a polarized order (O, µ). This will automatically yield Theorem 2.3. In fact, in Propositions 3.4 and 3.8, we will be able to conclude a rather more precise statement than the one given in Section 2.
The next proposition shows that the situation is simplified considerably in the non-twisting case.
If (O, µ) is a non twisting polarized order, then k O = k S for any totally real quadratic order S ⊂ O over R F and Proof. It is clear from definition 3.2 that the groups of twisting involutions V 0 (S) are trivial for any subring S of O. Since Gal(k O /k S ) ⊆ V 0 (S), this yields the first part of the proposition. As for the second, since Gal( which is a 2-torsion abelian finite group. Our claim now follows from the following lemma, which holds true for arbitrary pairs (O, µ). ✷ Proof. Let us identify F (µ) and F ( √ −D) through any fixed isomorphism. As Observe further that, if ξ f ∈ Ω is a root of unity of odd order f ≥ 3, then . This is a contradiction since DR F = ℘ 1 · ... · ℘ 2r , r > 0.
Finally, it is also impossible that there should exist ω ∈ F ( √ −D), ω 2 = λξ, ξ f = 1, f = 2 N f 0 with N ≥ 1 and f 0 ≥ 3 odd. Indeed, since in this case ξ ′ = ξ 2 N ∈ F ( √ −D) is a primitive root of unity of order f 0 , ω ′ = ξ f 0 +1 2 satisfies ω ′ 2 = ξ ′ . Then we would have ω ′ ω 2 = (ξ 2 N−1 )ξ and this would mean that [ ω ′ ω ] = [ω] ∈H, which is again a contradiction. This shows thatH is trivial and therefore H = In order to conclude the lemma, we only need to observe that both µ and ξ f +1 2 f ∈ F (µ) normalize the maximal order O for any odd f , because their respective reduced norms divide the discriminant D. ✷ Corollary 3.6. Let (O, µ) be a non-twisting polarized order and assume that F ( √ −D) is a CM-field with no purely imaginary roots of unity. Then, for any real quadratic order S over R F , k O /k R F = k S /k R F is at most a quadratic extension.
If, in addition, k R F admits a real embedding, then k O is a totally imaginary quadratic extension of k R F .
Proof. The first part follows directly from the above. As for the second, it follows from a theorem of Shimura [18] which asserts that the Shimura varieties X O,µ fail to have real points and hence the fields k O are purely imaginary. ✷ However, if on the other hand (O, µ) is twisting, the situation is more subtle and less homogenous as we now show.
This produces a natural action of U 0 on the set of twisting involutions of (O, µ) which is free simply because B is a division algebra. In order to show that it is transitive, let χ 1 , χ 2 be two twists.
is its own commutator subalgebra of B; further ω ∈ N B * (O) because its reduced norm is supported at the ramifying prime ideals ℘ | disc(B). Let us remark that, in the same way, We are now in a position to prove the lemma. Let ν ∈ V 0 be a fixed twisting involution. Then U 0 ⊂ V 0 : for any ω ∈ U 0 we have already shown that ων is again a twisting involution and hence (ων)ν = ω ∈ V 0 because V 0 is a 2-torsion abelian group. In addition, the above discussion shows that any element of V 0 either belongs to U 0 or is a twisting involution and that there is a non-canonical isomorphism V 0 /U 0 ≃ U 0 . ✷ Observe that in the twisting case, by the above lemma, U 0 acts freely and transitively on the set of twisting involutions of W 1 with respect to (O, µ). If (O, µ) is a twisting polarized order, let χ 1 , ..., χ s 0 ∈ O be representatives of the twists of (O, µ) up to multiplication by elements in F * . Then, (i) For any real quadratic order S, S ⊂ F (χ i ), 1 ≤ i ≤ s 0 ,

Proof.
If S ⊂ F (χ i ) for any i = 1, ..., s 0 , then V 0 (S) is trivial and hence, since Gal(k O /k S ) ⊆ V 0 (S), Gal(k O /k S ) is also trivial. If, on the other hand, S ⊆ F (χ i ) ∩ O, then V 0 (S) ≃ C 2 is generated by the twisting involution associated to χ i . Again, we deduce that in this case k O /k S is at most a quadratic extension.
With regard to the last statement, note that U 0 ⊇ [µ] is at least of order 2. Thus, if (O, µ) is a twisting polarized order, it follows from Lemma 3.4 that there exist two non-commuting twists χ, χ ′ ∈ O. Then R F [χ, χ ′ ] is a suborder of O and, since they both generate B over Q, the fields of moduli k O and k R F [χ,χ ′ ] are the same. This shows that k O ⊆ k S 1 · ... · k Ss 0 . The converse inclusion is obvious.
Finally, we deduce that k O /k R F is a (2, ..., 2)-extension of degree at most 2 2ω odd from Lemma 3.7. ✷ Remark 3.9. In the twisting case, the field of moduli of quaternionic multiplication is already generated by the field of moduli of any maximal real commutative multiplication but for finitely many exceptional cases. This homogeny does not occur in the non-twisting case.
In view of corollaries 3.4 and 3.8, the shape of the fields of moduli of the endomorphisms of the polarized abelian variety (A, L) differs considerably depending on whether it gives rise to a twisting polarized order (O, µ) or not.
For a maximal order O in a totally indefinite quaternion algebra B of principal reduced discriminant D ∈ F * + , it is then obvious to ask the questions whether (i) There exists µ ∈ O, µ 2 + D = 0 such that (O, µ) is twisting and, (ii) If (O, µ) is twisting, what is its twisting group V 0 . Both questions are particular instances of the ones considered in the appendix at the end of the article.

Fields of moduli versus fields of definition
In dimension 2, the results of the previous sections are particularly neat and can be made more complete. Let C/Q be a smooth irreducible curve of genus 2 and let (J(C), Θ C ) denote its principally polarized Jacobian variety. Assume that End Q (J(C)) = O is a maximal order in an (indefinite) quaternion algebra B over Q of reduced discriminant D = p 1 · ... · p 2r . Recall that O is unique up to conjugation or, equivalently by the Skolem-Noether Theorem, up to isomorphism. In the rational case, the Atkin-Lehner and the positive Atkin-Lehner groups coincide and If (O, µ) is a non twisting polarized order, then the field of moduli of quaternionic multiplication k O is at most a quadratic extension over the field of moduli k C of the curve C by In [11], Mestre studied the relation between the field of moduli k C = k J(C),Θ C of a curve of genus 2 and its possible fields of definition, under the sole hypothesis that the hyperelliptic involution is the only automorphism on the curve. Mestre constructed an obstruction in Br 2 (k C ) for C to be defined over its field of moduli. On identifying this obstruction with a quaternion algebra H C over k C , he showed that C admits a model over a number field K, k C ⊆ K, if and only if H C ⊗ K ≃ M 2 (K).
If Aut(C) C 2 , Cardona [1] has recently proved that C always admits a model over its field of moduli k C .
Assume now, as in the theorem above, that End Q (J(C)) ≃ O is a maximal order in a quaternion division algebra B over Q. Let K be a field of definition of C; note that, since End Q (J(C)) ⊗ Q = B is division, Aut(C) ≃ C 2 and therefore k C does not need to be a possible field of definition of the curve. Having made the choice of a model C/K, there is a minimal (Galois) field extension L/K of K such that End L (J(C)) ≃ O. This gives rise to a diagram of Galois extensions The nature of the Galois extensions L/K was studied in [4] and [5], while the relation between the field of moduli k O and the possible fields of definition L of the quaternionic multiplication was investigated by Jordan in [10]. In the next proposition we recall some of these facts, and we prove that L is the compositum of K and the field of moduli k O .

Proposition 4.2. 1
Let C/K be a smooth curve of genus 2 over a number field K and assume that End Q (J(C)) is a maximal quaternionic order O. Let L/K the minimal extension of K over which all endomorphisms of J(C) are defined. Then Proof. Statement (i) was proved in [4]. As for (ii), let M be any number field. Jordan proved in [10] that the pair (J(C), End(J(C))) admits a model over M if and only if M contains k O and M splits B. Since A is defined over K and all its endomorphisms are defined over L, we obtain that L ⊇ k O · K and B ⊗ Q L ≃ M 2 (L).
By the Skolem-Noether theorem, there exists α ∈ O such that αγ = −γα. For such an element α, we obtain from the above equality that φγ = −γφ. Since we also know that φ ∈ Q(γ), it follows that φ ∈ Q · γ. This enters in contradiction with the fact that φ is an automorphism. ✷ Example 4.3. Let C be the smooth projective curve of hyperelliptic model Let A = J(C)/K be the Jacobian variety of C over K = Q( √ −3). By [9], A is an abelian surface with quaternionic multiplication by a maximal order in the quaternion algebra of discriminant 10 2 . As it is explicitly shown in [9], there is an isomorphism between C and the conjugated curve C τ over Q. Hence, the field of moduli k C = Q is the field of rational numbers. By applying the algorithm proposed by Mestre in [11], we also obtain that the obstruction H C for C to admit a model over Q is not trivial.
is a minimal field of definition for C, though C also admits a model over any other quadratic field K ′ that splits H C .
In addition, it was shown in [4] that L = Q( √ −3, √ −11)/K is the minimal field of definition of the quaternionic endomorphisms of A. By a result of Shimura, Shimura curves fail to have rational points over real fields. Hence, k O must be a subfield of L that does not admit a real embedding. By Proposition 4.2, L = k O ( √ −3) and thus k O is either Q( √ −11) or L itself. It would be interesting to determine which of these two fields is k O . 3

Appendix: Integral quaternion basis and distance ideals
A quaternion algebra over a field F is a central simple algebra B over F of rank F (B) = 4. However, there are several classical and more explicit ways to describe them which we now review. Indeed, if L is a quadratic separable algebra over the field F and m ∈ F * is any non zero element, then the algebra B = L + Le with e 2 = m and eβ = β σ e for any β ∈ L, where σ denotes the non-trivial involution on L, is a quaternion algebra over F . The classical notation for it is B = (L, m). Conversely, any quaternion algebra over F is of this form ( [21]).
In addition, if char(F ) = 2, then with ij = −ji and i 2 = a ∈ F , j 2 = b ∈ F for any two elements a, b ∈ F * is again a quaternion algebra over F and again any quaternion algebra admits such a description. Note that the constructions are related since B = ( a,b F ) = (F (i), b). On a quaternion algebra B there is a canonical anti-involution β →β which is characterized by the fact that, when restricted to any embedded quadratic subalgebra L ⊂ B over F , it coincides with the non-trivial F -automorphism of L. Thus, if B = (L, m), thenβ = β 1 + β 2 e = β 1 σ − β 2 σ e. The reduced trace and norm on B are defined by tr(β) = β +β and n(β) = ββ.
Assume that F is either a global or a local field of char(F ) = 2 and let it be the field of fractions of a Dedekind domain R F . An order O in a quaternion algebra B is an R F -finitely generated subring such that O · F = B. Elements β ∈ O are roots of the monic polynomial x 2 − tr(β)x + n(β), tr(β), n(β) ∈ R F . We are now able to formulate the following question.
Question: Let B be a quaternion algebra over a global or local field F , char (F ) = 2, and let O be an order in B.
(2) If B ≃ (L, m) for a quadratic separable algebra over F and m ∈ R F , can one find χ ∈ O such that χ 2 = m, χβ =βχ for any β ∈ L?
We note that Question 2 may be considered as a refinement of Question 1. Indeed, let O be an order in B = ( a,b F ) and fix an arbitrary element i ∈ O such that i 2 = a. Then, while Question 1 asks whether there exist arbitrary elements ι, η ∈ O such that ι 2 = a, η 2 = b and ιη = −ηι, Question 2 wonders whether such an integral basis exists with η = i.
Obviously, Question 1 is answered positively whenever γ −1 Oγ ⊇ O 0 for some γ ∈ B * . The following proposition asserts that this is actually a necessary condition. Although it is not stated in this form in [3], it is due to Chinburg and Friedman, and follows from the ideas therein. It is a consequence of Hilbert's Satz 90. Let us agree to say that two An order O in B contains a basis ι, η ∈ O, ι 2 = a, η 2 = b, ιη = −ηι of B if, and only if, the type of O 0 is contained in the type of O.
By Hilbert's Satz 90, there exists ω ∈ F (η) such that ιi −1 = ωω −1 , that is, ι = ωω −1 i. Stated in this form, we need to find an element γ ∈ F (η) with γ −1 ωω −1 iγ = i. Since γi = iγ, we can choose γ = ω. ✷ An order O in B is maximal if it is not properly contained in any other. It is an Eichler order if it is the intersection of two maximal orders. The reduced discriminant ideal of an Eichler order is disc(O) = disc(B) · N for some integral ideal N of F , the level of O, coprime to disc(B) (see [21], p. 39). With this notation, maximal orders are Eichler orders of level 1. Proof. By [21], §2, there is only one type of Eichler orders of fixed level N in Since disc(O 0 ) = 4ab, as one can check, a necessary and sufficient condition on O to contain a conjugate order of O 0 is that N | 4ab. The corollary follows from proposition 5.1. ✷ In the global case, the approach to Question 1 can be made more effective under the assumption that B satisfies the Eichler condition. Namely, suppose that some archimedean place v of F does not ramify in B, that is, The following theorem of Eichler describes the set T (N ) of types of Eichler orders of given level N purely in terms of the arithmetic of F . Let Pic + (F ) be the narrow class group of F of fractional ideals up to principal fractional ideals (a) generated by elements a ∈ F * such that a > 0 at any real archimedean place v that ramifies in B and let h + (F ) = |Pic + (F )|.
Definition 5.3. The group Pic N + (F ) is the quotient of Pic + (F ) by the subgroup generated by the squares of fractional ideals of F , the prime ideals ℘ that ramify in B and the prime ideals q such that N has odd q-valuation.  η ∈ O such that η 2 = a. Then, it follows from the above proposition that Question 2 for (O, η) is answered in the affirmative for elements η lying on exactly f of the conjugation classes in E(a). Again, the cardinality of E(a) can be explicitly computed in many cases in terms of class numbers by means of the theory of Eichler optimal embeddings (cf. [21]).