Fields of definition of elliptic k -curves and the realizability of all genus 2 Sato--Tate groups over a number field
aa r X i v : . [ m a t h . N T ] F e b FIELDS OF DEFINITION OF ELLIPTIC k -CURVES AND THEREALIZABILITY OF ALL GENUS 2 SATO–TATE GROUPSOVER A NUMBER FIELD FRANCESC FIT´E AND XAVIER GUITART
Abstract.
Let A/ Q be an abelian variety of dimension g ≥ Q to E g , where E is an elliptic curve. If E does not have complexmultiplication (CM), by results of Ribet and Elkies concerning fields of defini-tion of elliptic Q -curves E is isogenous to a curve defined over a polyquadraticextension of Q . We show that one can adapt Ribet’s methods to study thefield of definition of E up to isogeny also in the CM case. We find two ap-plications of this analysis to the theory of Sato–Tate groups: First, we showthat 18 of the 34 possible Sato–Tate groups of abelian surfaces over Q occuramong at most 51 Q -isogeny classes of abelian surfaces over Q ; Second, we givea positive answer to a question of Serre concerning the existence of a numberfield over which abelian surfaces can be found realizing each of the 52 possibleSato–Tate groups of abelian surfaces. Contents
1. Introduction 12. Fields of definition of elliptic k -curves 52.1. Preliminary results on k -varieties 52.2. Powers of CM elliptic k -curves 83. The case of dimension g = 2 123.1. Galois module structures: group representation obstructions 143.2. Restriction of scalars: cohomological obstructions 163.3. Abelian surfaces over Q K/k ) ≃ S and M = Q ( √−
2) 254. Two applications to Sato–Tate groups 294.1. A finiteness result 294.2. A number field for all genus 2 Sato-Tate groups 30References 351.
Introduction
It is well known that there exist three possibilities for the Sato–Tate group of anelliptic curve E defined over a number field k : the special unitary group SU(2) ofdegree 2, the unitary group U(1) of degree 1 embedded in SU(2), and the normalizer Date : September 12, 2018.Fit´e was funded by the German Research Council via SFB/TR 45.Guitart was partially funded by MTM2012-33830 and MTM2012-34611. k -CURVES AND GENUS 2 SATO–TATE GROUPS 2 N (U(1)) of U(1) in SU(2). These three possibilities are in accordance with thefollowing trichotomy: the elliptic curve does not have complex multiplication (CM),the elliptic curve has CM defined over k , and the elliptic curve has CM but notdefined over k .From this description, it is easy to see that there exists a number field (forexample, take any quadratic imaginary field of class number 1) over which threeelliptic curves can be defined realizing each of the three Sato–Tate groups SU(2),U(1), and N (U(1)).In [FKRS12], it was shown that there exist 52 possible Sato–Tate groups ofabelian surfaces over number fields, all of which occur for some choice of the num-ber field and of the abelian surface defined over it. Let N ST , ( k ) denote the num-ber of subgroups of USp(4) up to conjugacy that arise as Sato–Tate groups ofabelian surfaces defined over the number field k . In [FKRS12], it was proven that N ST , ( Q ) = 34. Serre has asked the following question. Question 1.1.
Does there exist a number field k over which abelian surfacescan be found realizing each of the possible Sato–Tate groups of abelian surfacesover number fields? In other words, does there exist a number field k such that N ST , ( k ) = 52 ? It is well-known that the connected component of the identity of the Sato–Tategroup is invariant under base change. However, its group of components is sensitiveto base change, and it is therefore not necessarily true (and in fact, it is not) that thecompositum of all fields of definition of the examples in [FKRS12] gives a positiveanswer to the above question.The question is thus on whether the conditions imposed on k by each of thepossible groups of components are compatible or not among them. Fundamental tothis analysis is the existence of an isomorphism between the group of componentsof the Sato–Tate group of an abelian surface A defined over k and the Galois groupGal( K/k ), where
K/k is the minimal extension over which all of the endomorphismsof A are defined.Let us look at a concrete example of the type of conditions on k imposed bycertain groups of components. We will consider abelian surfaces A defined over anumber field k satisfying(P) Gal( K/k ) contains S .One can easily show that (P) implies that A is isogenous over Q to the square ofan elliptic curve with CM, say by a quadratic imaginary field M . There are onlythree Sato–Tate groups (denoted by O , O , and J ( O ) in [FKRS12]) whose group ofcomponents contains the symmetric group S . The dictionary between Sato–Tategroups and Galois endomorphism types of [FKRS12] ensures that O only arisesamong A satisfying (P) for which M ⊆ k , whereas O and J ( O ) can occur only for A satisfying (P) with M k .Up to our knowledge, all the examples in the existing literature of abelian sur-faces A satisfying (P) have M = Q ( √− A with M = Q ( √− A satisfying (P)with M = Q ( √− M = Q ( √−
2) might be a necessarycondition for A to satisfy (P). If this was the case, then Question 1.1 would have a LLIPTIC k -CURVES AND GENUS 2 SATO–TATE GROUPS 3 negative answer, since in order to realize, for example, both O and J ( O ) the field k would be forced to contain and not contain Q ( √−
2) simultaneously.As a consequence of the preceding discussion, one is naturally led by Question 1.1to the following question regarding a basic aspect of the arithmetic of abelianvarieties over Q . Question 1.2.
Let A be an abelian variety defined over Q of dimension g ≥ thatis Q -isogenous to E g , where E is an elliptic curve over Q with CM by M . (A) Which is the set of possibilities for M ? (B) Let
K/M be the minimal extension over which all the endomorphisms of A are defined. Does the prescription of Gal(
K/M ) impose further restrictionson M ? If g = 1, then the theory of complex multiplication shows that M is a quadraticimaginary field of class number 1 and thus there are only 9 possibilities for M . Inthis work, we provide an answer to Question 1.2 for g = 2, which sets the basis onwhich we build a positive answer to Question 1.1. Main results.
As the case g = 1 may suggest, an answer to Question 1.2 willfollow, via the theory of complex multiplication, from gaining control on the fieldof definition of the elliptic factor E . The study of the field of definition of E willbe carried out in § Theorem 1.3.
Let E be an elliptic curve defined over Q with CM by a quadraticimaginary field M such that E g , for some g ≥ , is Q -isogenous to an abelianvariety A defined over k . Assume that M is contained in k . Let K/k be theminimal extension over which all the endomorphisms of A are defined. Then E is Q -isogenous to an elliptic curve E ∗ defined over an abelian subextension F/k of K/k such that every element in
Gal(
F/k ) has order dividing g . Observe that the elliptic curve E in the preceding theorem is an elliptic k -curvein the sense of Ribet (the notion of elliptic k -curve and, more generally, of abelian k -variety, will be recalled in § k -curve C without CM (achieved by Ribet [Rib94] and Elkies [Elk04]by means of completely different methods) is well known: C admits a model upto isogeny defined over a polyquadratic extension of k . We will show that in ourparticular situation Ribet’s methods can also be applied to the CM case. Thecrucial idea is that E g admitting a model over k implies that the g th power of thecohomology class γ E ∈ H ( G k , M ∗ ) naturally attached to E is trivial, a propertythat does not necessarily hold for an arbitrary CM elliptic k -curve C .Combining Theorem 1.3 with bounds on the degree of K/k (deduced from themethods of [Sil92] and [GK16]), one obtains the following answer to part ( A ) ofQuestion 1.2. Theorem 1.4.
With the notations as in Question 1.2, the class group
Cl( M ) of M has exponent dividing g . If moreover g is prime, then Cl( M ) = ( , C g , or C g × C g if g = 2 , or C g otherwise.In particular, if g = 2 , there are at most possibilities for M . That is, the composition of a finite number of quadratic extensions.
LLIPTIC k -CURVES AND GENUS 2 SATO–TATE GROUPS 4 We refer the reader to Theorem 3.21 for a more precise version of the abovetheorem in the case g = 2, which accounts also for part ( B ) of Question 1.2. Justto show the flavour of the result, let us state a particular instance of it: if for exampleGal( K/M ) ≃ S , then M is either Q ( √−
2) or a quadratic imaginary field of classnumber 2 and distinct from Q ( √− Q ( √− Q ( √− Q ( √− M are obtained by means of two different methods.First, we consider obstructions coming from group representation theory. Theseare obtained by pushing a bit further the study in [FS14, §
3] of the Galois modulestructure of the ring of endomorphisms of an abelian surface that is Q -isogenous tothe square of an elliptic curve. This is done in § K/k E inthe case Gal( K/k ) ≃ S . We again benefit from a technique exploited in Ribet’swork: we look at this algebra as a twisted group algebra. A key step towards de-termining its structure is to compute its center; we devote § § § § g = 2. Theorem 1.5.
Among the possibilities for the Sato–Tate group of an abeliansurface defined over Q , the with identity component isomorphic to U(1) occuramong at most Q -isogeny classes of abelian surfaces over Q . We refer to Theorem 4.1 for a more precise version of the above statement. Atthis stage, we are ready to provide an affirmative answer to Question 1.1, which isthe main result of this paper (see Theorem 4.2).
Theorem 1.6.
One has N ST , ( k ) = 52 for k := Q ( √− , √− , √− , √− , √ · , √− . After the discussion following Question 1.1, it is clear that constructing anabelian surface defined over k satisfying (P) and for which M = Q ( √−
2) is acrucial step in the proof. We devote the whole of § A defined over k = Q ( √− Q -isogenous to the square of an elliptic curvewith CM by M = Q ( √− K/k ) ≃ S . This abelian surface A is obtained as a simple factor of the restriction of scalars of a certain elliptic curvewith CM by Q ( √−
40) over a suitable extension
K/k ; in the construction, we makecrucial use of the techniques developed in § §
4, requires less elaborate techniques,though still forces k to contain a few more additional quadratic fields. Notations and terminology.
Throughout this article, k will be a number fieldassumed to be contained in a fixed algebraic closure Q of Q . We will write G k forthe absolute Galois group Gal( Q /k ). All field extensions of k considered will beassumed to be algebraic and contained in Q . By ζ r we will denote a primitive r throot of unity, and g will be an integer ≥
1. We will work in the category of abelianvarieties up to isogeny over k . This means that isogenies become invertible in thiscategory and therefore, if A is an abelian variety defined over k , we will write End( A )to denote the Q -algebra of endomorphisms defined over k (what some authors writeEnd( A ) ⊗ Q or End ( A )). Similarly, if B is an abelian variety defined over k , thenHom( A, B ) will denote the Q -vector space of homomorphisms from A to B that LLIPTIC k -CURVES AND GENUS 2 SATO–TATE GROUPS 5 are defined over k . Although isogenies will be the isomorphisms of our workingcategory, we will still write A ∼ B to mean that A and B are isogenous over k . If L/k is a field extension, then A L will denote the base change of A from k to L , andwe will write A L ∼ B L if A and B become isogenous over L (in particular, we willwrite Hom( A L , B L ) to refer to what some authors write as Hom L ( A, B )). If C is anabelian variety over L , for short, we will say that C admits a model up to isogenyover k if there exists an abelian variety C defined over k such that C Q ∼ C , Q .Finally, if B is an algebra we will denote by Z ( B ) its center. Acknowledgments.
Fit´e is thankful to CIRM and ICERM for invitations inMay and in September of 2015, respectively, which facilitated fruitful conversa-tions with Drew Sutherland regarding this work. Thanks to Carlos de Vera, ElisaLorenzo, and Mark Watkins for helpful comments. We are thankful to Kiran Ked-laya and Jordi Quer for their inspiring suggestions, specially regarding the proof ofCorollary 2.17. 2.
Fields of definition of elliptic k -curves In this section we study fields of definition up to isogeny of certain ellipticcurves E with CM. Namely, those with the property that the CM field is con-tained in k and such that E g admits a model up to isogeny over k . The main toolthat we use is to exploit the fact that, in this setting, E is an elliptic k -curve.2.1. Preliminary results on k -varieties. In this section we collect some back-ground and preliminary results on abelian k -varieties. In fact, we are mainly inter-ested in the 1-dimensional case of elliptic k -curves. The reader can consult [Rib94],[Que00], and [Pyl02] as general references for k -curves and k -varieties. Definition . An abelian variety B/ Q is called an abelian k -variety(or just a k -variety, for short) if for all σ ∈ G k there exists an isogeny µ σ : σ B → B which is compatible with End( B ), in the sense that the diagram σ B σ ϕ (cid:15) (cid:15) µ σ / / B ϕ (cid:15) (cid:15) σ B µ σ / / B (2.1)commutes for all ϕ ∈ End( B ).A simple calculation shows that the property of being a k -variety only dependson the Q -isogeny class of B . In fact, if K is an extension of k one says that B is a k -variety defined over K if B is defined over K and B Q is a k -variety.Let B be a k -variety with endomorphism algebra B = End( B ), and let R = Z ( B )denote its center. The variety B has a model defined over a number field. Byenlarging this field if necessary, we can assume that it is also a field of definitionof the isogenies between B and its conjugates. Thus we can (and do) fix a systemof compatible isogenies { µ σ : σ B → B } σ ∈ G k which is locally constant. For every σ, τ ∈ G k define c B ( σ, τ ) = µ σ ❛ σ µ τ ❛ µ − στ . A short computation shows that c B ( σ, τ ) lies in R ∗ , and that the map σ, τ c B ( σ, τ ) LLIPTIC k -CURVES AND GENUS 2 SATO–TATE GROUPS 6 is a continuous 2-cocycle of G k with values in R ∗ (considered as a G k -module withtrivial action). Denote by γ B ∈ H ( G k , R ∗ ) the cohomology class of c B . To thebest of our knowledge this cohomology class was introduced by Ribet in [Rib92],and it is one of the main tools for studying the arithmetic of k -varieties. The nextproposition summarizes some of the main properties of c B and γ B . Proposition 2.2. i) If c ′ B is a cocycle cohomologous to c B , there exist compatibleisogenies λ σ : σ B → B such that c ′ B ( σ, τ ) = λ σ ❛ σ λ τ ❛ λ − στ .ii) The cohomology class γ B ∈ H ( G k , R ∗ ) only depends on the Q -isogeny classof B . More precisely, if β : A → B is an isogeny, then γ A = γ B , under theidentification of centers Z ( B ) ≃ Z (End( A )) provided by β .iii) For every n ∈ Z ≥ the isotypical variety B n is also a k -variety, and γ B = γ B n ,under the identification Z ( B ) ≃ Z ( B n ) .Proof. For i ), the cocycle c ′ B can be written as c ′ B ( σ, τ ) = c B ( σ, τ ) ❛ a σ ❛ a τ ❛ a − στ for some a σ , a τ , a στ ∈ R ∗ . Then, for every s ∈ G k the isogeny λ s = a s ❛ µ s iscompatible, because a s ∈ R ∗ , and it satisfies the required property.As for ii ) one can define isogenies λ σ : σ A → A as λ σ = β − ❛ µ σ ❛ σ β . It is easyto check that they are compatible and that c A ( σ, τ ) = β − ❛ c B ( σ, τ ) ❛ β .Finally, iii ) is proved by considering the isogenies µ ⊕ nσ : σ B n → B , which arecompatible and can be thus used to compute c B n ( σ, τ ), which is seen to coincidewith c n ⊕ B . (cid:3) Proposition 2.3.
Let B/ Q be a simple abelian k -variety and suppose that End( B ) has Schur index t . Suppose that there exists a variety A/k such that A Q ∼ B n .Then γ B lies in H ( G k , R ∗ )[ nt ] ; that is, ( γ B ) nt = 1 .Proof. Let { µ σ : σ B → B } σ ∈ G k be a compatible system of isogenies and let c B ( σ, τ ) = µ σ ❛ σ µ τ ❛ µ − στ be the corresponding cocycle. By Proposition 2.2 ii ) we have that A Q is a k -variety as well, and therefore there exists a compatible system of isogenies α σ : σ A → A such that c A ( σ, τ ) = α σ ❛ σ α τ ❛ α − στ . (2.2)Since A is defined over k , we see that α s lies in End( A Q ) for every s ∈ G k . Inparticular, the compatibility condition applied to the endomorphism α τ then reads σ α τ = α − σ ❛ α τ ❛ α σ . Plugging this into (2.2) gives c A ( σ, τ ) = α τ ❛ α σ ❛ α − στ . Now by Proposition 2.2 ii ) we have that γ A = γ B n ; by iii ) we also see that γ A = γ B and by i ) we can in fact assume that the cocycles are equal: c B ( σ, τ ) = c A ( σ, τ ) = α τ ❛ α σ ❛ α − στ , (2.3)where the term c B ( σ, τ ) is seen as lying in End( A Q ) by means of the identification R ≃ Z (End( A Q )). In turn, we can interpret (2.3) as an equality in M n ( B ) thanksto the isomorphism End( A Q ) ≃ M n ( B ). Now we take reduced norms and we usethe fact that c B ( σ, τ ) lies in F , so that its reduced norm equals c B ( σ, τ ) nt . Thuswe obtain: c B ( σ, τ ) nt = nr( α τ ) ❛ nr( α σ ) ❛ nr( α στ ) − . This expresses c ntB as the coboundary of the map s nr( α s ) ∈ R ∗ . (cid:3) LLIPTIC k -CURVES AND GENUS 2 SATO–TATE GROUPS 7 Corollary 2.4.
Suppose that
A/k is an abelian variety such that A Q ∼ C g , forsome k -curve C/ Q . Then γ C lies in H ( G k , R ∗ )[ g ] .Proof. It follows directly from the fact that End( C ) has Schur index 1. (cid:3) Remark . Ribet showed [Rib94, Prop. 3.2] that if C is a k -curve without CMthen γ C has order dividing 2. This is not true in general for CM k -curves. Thepoint of Corollary 2.4 is that an analogous statement holds in the case that C hasCM and C g admits a model up to isogeny defined over k . Proposition 2.6.
Let L ⊂ Q be an extension of k and let A/L be a k -variety.Then Z (End( A Q )) ⊆ End( A ) .Proof. For all σ ∈ G L we have that σ A = A . Then the isogeny µ σ : σ A → A canbe identified with an element of End( A ). For any ϕ ∈ Z (End( A Q )) we have that σ ϕ = µ − σ ❛ ϕ ❛ µ σ = ϕ . (cid:3) The main examples of k -varieties that we will consider in this note are CM ellipticcurves whose field of CM is contained in k . The following result is well-known (itfollows, for instance, from [Sil94, Thm. 2.2]). Proposition 2.7.
Let
E/L be an elliptic curve with CM by an imaginary quadraticfield M . If L contains M , then E is an M -curve. In particular, it is a k -curve forany k containing M . For further reference, we record the following consequence of the results we haveseen so far.
Proposition 2.8.
Let A be an abelian surface defined over a number field k . Sup-pose that A Q ∼ E , where E is an elliptic curve with CM by M , and assume that M ⊆ k . Then Z (End( A Q )) ⊆ End( A ) .Proof. Since E is a CM elliptic curve defined over an extension of k and M ⊆ k we see that E is a k -curve. Now by Proposition 2.2 A is also a k -variety, and theresult is then a direct consequence of Proposition 2.6. (cid:3) We will use the notion of k -varieties completely defined over a field, a terminologywhich was introduced in [Que00, p. 2]. Definition . Let B be a k -variety defined over a number field K . One says that B is completely defined over K if there exist compatible isogenies { µ σ } σ ∈ G k that aredefined over K .If B is completely defined over K the map c KB : Gal( K/k ) × Gal(
K/k ) −→ R ∗ , c KB ( σ, τ ) = µ σ ❛ σ µ τ ❛ µ − στ (2.4)is a two-cocycle and its cohomology class γ KB ∈ H (Gal( K/k ) , R ∗ ) only depends onthe K -isogeny class of B . Remark . It follows from the definitions that the image of γ KB under the in-flation map H (Gal( K/k ) , R ∗ ) → H ( G k , R ∗ ) is precisely the cohomology class γ B defined earlier. In many applications, for instance when one is interested instudying properties of the K -isogeny class of B , it is important to work with thecohomology class γ KB . LLIPTIC k -CURVES AND GENUS 2 SATO–TATE GROUPS 8 Powers of CM elliptic k -curves. Throughout this section, let E be anelliptic curve defined over Q with CM by a quadratic imaginary field M such that E g is Q -isogenous to an abelian variety A defined over k . Assume that M is containedin k . Note that, in view of Proposition 2.7, E is a k -curve.Let K/k denote the smallest extension such that End( A K ) = End( A Q ). Themain result of this section is Theorem 2.14, which ensures the existence of anabelian subextension of K/k with Galois group killed by g over which E admits amodel up to isogeny. Remark . Ribet [Rib94] and Elkies [Elk04] have given two alternative proofsof the fact that any k -curve without CM admits a model up to isogeny definedover a polyquadratic extension of k . Theorem 2.14 may be seen as a extension ofRibet’s and Elkies’ result to the case in which E has CM, but under the restrictiveassumption that E g admits a model up to isogeny over k .Before proceeding to the proof of Theorem 2.14, we introduce some notation andwe recall Weil’s descent criterion, a fundamental result in our arguments.Note that the definition of K/k implies that A K ∼ E g , where E is an ellipticcurve defined over K . Since σ E g ∼ σ A K ∼ A K ∼ E g , for σ ∈ G k , by Poincar´e’s decomposition theorem there exists an isogeny µ σ : σ E → E defined over K . Note that E is a model of E up to isogeny defined over K .Even more, according to the terminology of Definition 2.9, E is an elliptic k -curvecompletely defined over K . Define the 2-cocycle c E : G k × G k → M ∗ , c E ( σ, τ ) = µ σ ❛ σ µ τ ❛ µ − στ , and let γ = γ E denote its cohomology class in H ( G k , M ∗ ). By Remark 2.10, theclass γ lies in the image of the inflation mapInf : H (Gal( K/k ) , M ∗ ) −→ H ( G k , M ∗ ) . The following result [Rib92, Thm. 8.2], written in the spirit of [Rib94, Prop. 3.1],is crucial for our purposes.
Proposition 2.12 (Weil’s descent criterion) . Let
F/k be an algebraic extension.If γ lies in the kernel of the restriction map H ( G k , M ∗ ) → H ( G F , M ∗ ) , then E admits a model up to isogeny defined over F .Proof. Of course it is enough to prove that γ = 1, implies that E admits a modelup to isogeny defined over k . But γ = 1 means that there is a locally constantfunction d : G k → M ∗ such that c E ( στ ) = d ( σ ) d ( τ ) /d ( στ ). Then the isogenies λ σ := 1 /d ( σ ) ❛ µ σ satisfy λ σ ❛ σ λ τ ❛ λ − στ = 1, and then one concludes by applying[Rib92, Thm. 8.2]. (cid:3) Remark . Let γ K ∈ H (Gal( K/k ) , M ∗ ) be the cohomology class constructedas in (2.4). Since γ K is constructed in terms of isogenies that are defined over K , LLIPTIC k -CURVES AND GENUS 2 SATO–TATE GROUPS 9 the argument above can be adapted to show that if F/k is subextension of
K/k and γ K lies in the kernel of the restriction map H (Gal( K/k ) , M ∗ ) −→ H (Gal( K/F ) , M ∗ ) , then E is K -isogenous to an elliptic curve defined over F . Theorem 2.14.
Let E be an elliptic curve defined over Q with CM by a quadraticimaginary field M such that E g , for g ≥ , is Q -isogenous to an abelian variety A defined over k . Assume that M is contained in k . Let H denote the Hilbert classfield of M and write F = Hk . Then, E admits a model up to isogeny definedover the abelian subextension F/k of K/k and every element of
Gal(
F/k ) has orderdividing g .Proof. Of course, it is enough to prove the equivalent statement for E (where E is attached to E as above). We will follow the strategy of [Rib94, Thm. 3.3].Although, Ribet’s result is stated under different hypotheses, we will show that γ = γ E lying in H ( G k , M ∗ )[ g ] is essentially all that is needed to run his argument.Let U denote the group of roots of unity of M and write P := M ∗ /U . As in [Rib94,Lemma 3.5], it follows from Dirichlet’s unit theorem that P is a free abelian group.Therefore, the exact sequence1 → U → M ∗ → P → H ( G k , M ∗ )[ g ] ≃ H ( G k , U )[ g ] × Hom( G k , P/P g ) . Indeed, the long exact cohomology sequence associated to the exact sequence1 → P x x g −→ P → P/P g → , together with the freeness of P , implies that we have an isomorphism H ( G k , P )[ g ] ≃ Hom( G k , P/P g ) . Note that by Corollary 2.4, the class γ belongs to H ( G k , M ∗ )[ g ]. Let γ U (resp. γ )be the projection of γ onto the first (resp. second) factor of the decomposition (2.5).Since γ is the inflation of a class in H (Gal( K/k ) , M ∗ ), we see that the component γ lies in the image of the inflation mapInf : Hom(Gal( K/k ) , P/P g ) −→ Hom( G k , P/P g ) . In particular, if we let L be the subfield of K fixed by the subgroup of Gal( K/k )generated by the g -th powers of the elements of Gal( K/k ), it is clear that therestriction of γ to G L is trivial. But now, by Lemma 2.15 below, γ is trivial over G L and thus, by Proposition 2.12, E admits a model E over L up to isogeny. Let O c be the order, say of conductor c , by which E has CM, and let H c = M ( j ( E ))denote the ray class field of conductor c of M . Since E is defined over L , we have(2.6) kH c = k ( j ( E )) ⊆ L. Let O be the maximal order of M and let E ∗ be an elliptic curve with CM by O defined over F = kH . Note that E ∗ is Q -isogenous to E . We have(2.7) F = k ( j ( E ∗ )) = kH ⊆ kH c . More explicitly, U = {± } , unless E has CM by M = Q ( √−
1) or Q ( √− U = h ζ i or h ζ i , respectively. LLIPTIC k -CURVES AND GENUS 2 SATO–TATE GROUPS 10 The theorem now follows from (2.6), (2.7), and the fact that Gal(
L/k ) is killedby g . (cid:3) Lemma 2.15.
With the notations as in the proof of Theorem 2.14, if γ trivializesover a subextension L/k of K/k , then so does γ U . That is, if γ has trivial restrictionto H ( G L , M ∗ ) , then so does γ U .Proof. It is enough to prove that if γ = 1 then γ U = 1. To show this, we will followRibet again. As proven in [Rib94, § H ( G k , ( M ⊗ Q Q ) ∗ ) = 1 , H ( G k , ( M ⊗ Q Q ) ∗ ) ≃ H ( G k , Q ∗ ) ⊕ H ( G k , Q ∗ ) , where G k acts trivially on M and via the natural Galois action on Q .Then, the exact sequence1 → M ∗ → ( M ⊗ Q Q ) ∗ → ( M ⊗ Q Q ) ∗ /M ∗ → , yields the following long exact sequence in cohomology → H ( G k , ( M ⊗ Q Q ) ∗ /M ∗ ) δ → H ( G k , M ∗ ) → H ( G k , Q ∗ ) ⊕ H ( G k , Q ∗ ) → · · · . As in [Rib94, Lem. 4.3], one finds that γ lies in the image of δ . Indeed, in our case M ⊗ Q ≃ Q ⊕ Q as G k -modules. Fix an invariant differential ω of E . For every σ ∈ G k there is a unique λ σ ∈ Q characterized by the identity µ ∗ σ ( ω ) = λ σ · σ ω .Then the map σ ( λ σ , λ σ ), where the bar stands for complex conjugation, givesrise to a cohomology class in H ( G k , ( M ⊗ Q ) ∗ /M ), whose image under δ coincideswith γ .Since γ is trivial, this means that γ U lies in the kernel of the map j : H ( G k , U )[ g ] → H ( G k , Q ∗ ) ⊕ H ( G k , Q ∗ ) . Observe that U is contained in k so that the trivial action of G k on U coincides withthe natural Galois action. Since H ( G k , U ) ≃ H ( G k , Q ∗ )[ n ] (here Q is endowedwith the natural Galois action of G k , and n is the cardinality of U ), we see that j is injective and thus γ U must be trivial. (cid:3) Corollary 2.16.
Let A be an abelian variety defined over Q of dimension g thatis Q -isogenous to E g , where E is an elliptic curve over Q with CM by M . Thenevery element in Cl( M ) has order dividing g .Proof. Applying Theorem 2.14 to the base change A M of A to M , we deduce thatevery element in Gal( H/M ) has order dividing g , where H stands for the Hilbertclass field of M . The corollary follows from the fact that Gal( H/M ) is isomorphicto Cl( M ). (cid:3) Corollary 2.17.
Let A be an abelian variety defined over Q that is Q -isogenousto E g , where E is an elliptic curve defined over Q with CM by a quadratic imaginaryfield M and g is a prime number. Then Cl( M ) = ( , C g , or C g × C g if g = 2 , or C g otherwise.In particular, if g = 2 , there are at most possibilities for M . LLIPTIC k -CURVES AND GENUS 2 SATO–TATE GROUPS 11 Proof.
By Corollary 2.16, the group Cl( M ) is a quotient of Gal( K/M ) of the formC gr , for some r ≥
0. For g = 2, the possibilities for Gal( K/M ) are well known (seeRemark 3.1 in the next section, for example) and it turns out that r ≤
2. It is wellknown that the number of quadratic imaginary fields with class group isomorphicto the trivial group, C , C × C are respectively 9, 18, 24 (see Table 3 for thecomplete list).Suppose now that g ≥
3. For p a prime number, define r ( g, p ) := X j ≥ (cid:22) gp j ( p − (cid:23) . (2.8)By [Sil92, Thm. 4.1], the maximal power of p dividing the order of Gal( K/M ) is r ( g, p ). Guralnick and Kedlaya [GK16, Thm. 1.1] have shown that the same holdstrue if one replaces r ( g, p ) by r ′ ( g, p ) := r ( g, p ) − g − p = 2,max { r ( g, p ) − , } if p is a Fermat prime, r ( g, p ) otherwise.This bound is sharp in the context of general abelian varieties (see [GK16, Thm.6.3]). However, if A is as in the statement of the corollary (that is, geometricallyisogenous to the power of an elliptic curve with CM), one can do better. Indeed, onecan specialize the proof of [GK16, Thm. 5.4] to the particular case in which V ≃ W g as G -modules, as in our setting A K ∼ E g for an elliptic curve E defined over K .Therefore End G ( W ) = M , End G ( V ) = M g ( M ), and Z (End G ( W )) = M . In thenotations of [GK16, Thm. 5.4], this is yields the values a = 2 , b = 1 , c = g . Then, one deduces that the maximal power of g dividing the order of Gal( K/M ) isbounded by r (GL( bc ) M , g ) = r (GL( g ) M , g ) := m ( M, g ) (cid:22) gt ( M, g ) (cid:23) + (cid:22) t ( M, g ) (cid:23) , where, as defined in [GK16, Rmk. 3.6], we have m ( M, g ) := min { m ≥ | M ∩ Q ( µ g m ) = M ∩ Q ( µ g ∞ ) } and t ( M, g ) := [ Q ( µ g m ( M,g ) ) : M ∩ Q ( µ g m ( M,g ) )] . Since Q ( µ g m ( M,g ) ) ramifies only at g , we have that either t ( M, g ) = g m ( M,g ) − ( g − M = Q ( √− g ) and g ≡ r (GL( g ) M , g ) ≤ g = 2). In the second case, Cl( M ) is necessarilytrivial; indeed, by the reduction theory of positive definite binary quadratic forms(see for example [Cox89, Chap. I, § M ) ≤ g . (cid:3) Remark . There is a finite number of quadratic imaginary fields of fixed classnumber (see [Hei34]). Using Sage and Pari one can compute the number of imag-inary quadratic fields with class group C g , for 3 ≤ g ≤
97 prime, relying on thebounds for their fundamental discriminants provided by [Wat03, Table 4].
Remark . Also for g not necessarily prime, one can proceed as in the proof ofCorollary 2.17 to gain control on the size of Cl( M ). In general, one obtains thatCl( M ) has exponent dividing g and order dividing Q p | g p r ′ ( g,p ) . LLIPTIC k -CURVES AND GENUS 2 SATO–TATE GROUPS 12 Remark . One may wonder which of the cases allowed by Corollary 2.17 actuallyarise. By taking g -th powers (resp. Weil restrictions of scalars) of elliptic curvesit is clear that all quadratic imaginary fields of class number 1 (resp. with classgroup C g ) do appear. It would be an interesting problem to determine the exactset of possibilities for the M with Cl( M ) ≃ C × C for which an abelian surface A as in the corollary exists.The above corollary answers part ( A ) of Question 1.2. Providing an answer topart ( B ) of the question for g = 2 will be the goal of §
3. Before, for the sake ofcompleteness, we state a result analogous to Theorem 2.14 for the non-CM case.
Theorem 2.21.
Let E be an elliptic curve defined over Q without CM such that E g is Q -isogenous to an abelian variety A defined over k . Then E admits a model upto isogeny definedi) over k if g is odd; orii) over a polyquadratic subextension of K/k if g is even.Proof. Let γ denote the cohomology class attached to E . On the one hand, byProposition 2.3, the class γ g is trivial. On the other hand, [Rib94, Prop. 3.2]asserts that γ has order dividing 2. Proposition 2.12 implies then that i ) holds. It isstraightforward to adapt the proof of Theorem 2.14 to justify that the polyquadraticextension over which γ trivializes is in fact contained in K/k . (cid:3) The case of dimension g = 2In this section, E will denote an elliptic curve defined over Q with CM by aquadratic imaginary field M such that E is Q -isogenous to an abelian surface A defined over k . We assume that M is contained in k , so that E is a k -curve, andwe let K/k denote the smallest extension such that End( A K ) = End( A Q ).Our goal is to obtain a more precise version of Corollary 2.17. This will beachieved by Theorem 3.21. We will benefit from the fact that for g = 2, thestructure of Gal( K/k ) is very well understood.
Remark . It is well known that if M ⊆ k the possibilities for Gal( K/k ) are thecyclic group C n for n ∈ { , , , , } , the dihedral group D n for n ∈ { , , , } , thealternating group A , and the symmetric group S . There are at least two ways inwhich this can be deduced: first, by means of the analysis of the finite subgroupsof PGL ( M ), (see [Bea10], [CF00]); second, as a byproduct of the classification ofSato–Tate groups of abelian surfaces (see [FKRS12, § r ( A ) be defined in terms of Gal( K/k ) as specified on Table 1. In other words, r ( A ) is the maximum value of r for which there exists a Galois subextension F/k of K/k with Gal(
F/k ) ≃ C r . Proposition 3.2.
Let E be an elliptic curve defined over Q with CM by a quadraticimaginary field M such that E is Q -isogenous to an abelian surface A/k . Assumethat M ⊆ k . Then E admits a model up to isogeny E ∗ defined over a Galoissubextension F/k of K/k such that
Gal(
F/k ) ≃ C r ( A ) . The above proposition can certainly be deduced from Theorem 2.14 and Re-mark 3.1, but we wish to present a slightly different argument that renders, for g = 2, a shortcut in the proof. LLIPTIC k -CURVES AND GENUS 2 SATO–TATE GROUPS 13 r ( A ) Gal( K/k ) r ( A ) Gal( K/k )0 C Table 1.
The rank r ( A ) in terms of Gal( K/k ). Proof.
Recover decomposition (2.5), specialize it to g = 2, and resume with thefinal argument in the proof of [Rib94, Thm. 3.3]. If Br( k ) denotes the Brauergroup of k , then we have H ( G k , M ∗ )[2] ≃ Br( k )[2] × Hom( G k , P/P ) . If we let γ = γ E denote the cohomology class attached to E and ( γ U , γ ) its com-ponents under the above decomposition, then γ factors through a polyquadraticextension k ′ of k . Moreover, a theorem of Merkur’ev [Mer81] shows that Br( k )[2]is generated by the classes of quaternion algebras over k , and it is well known thateach quaternion algebra over k is split over a quadratic extension of k . Proposi-tion 2.12 thus tells us that E admits a model E up to isogeny over a polyquadraticextension k /k .Now let E /K be an elliptic curve such that A K ∼ E . We thus have that(3.1) Q ( j ( E )) ⊆ K and Q ( j ( E )) ⊆ k . We can now conclude in a similar manner as we did with the proof of Theorem 2.14.Let O be the maximal order of M and let E ∗ be an elliptic curve with CM by O .Both E and E are Q -isogenous to E ∗ . Let O i denote the order by which E i has CM, for i = 1 ,
2, and let c i be its conductor. Since the ray class field H c i = M ( j ( E i )) of conductor c i contains the Hilbert class field H = M ( j ( E ∗ )), we havethat M ( j ( E ∗ )) ⊆ M ( j ( E i )). It follows from (3.1) that F := k ( j ( E ∗ )) ⊆ K ∩ k . Since [ K ∩ k : k ] ≤ r ( A ) , we have that E ∗ F is a model up to isogeny of E enjoyingthe properties stated in the proposition. (cid:3) Remark . It follows from the above proof that any elliptic curve with CM bythe maximal order of M can be taken as the model E ∗ of Proposition 3.2, in whichcase F can be taken equal to k ( j ( E ∗ )). Remark . If E does not have CM, then the possibilities for Gal( K/k ) are those ofRemark 3.1 excluding A and S . Theorem 2.21 has then the following consequence.Let E be an elliptic curve without CM defined over Q such that E has a model upto isogeny defined over k . Then E admits a model up to isogeny E ∗ defined over abiquadratic subextension F/k of K/k .In the following § § M imposedby Gal( K/M ). In § LLIPTIC k -CURVES AND GENUS 2 SATO–TATE GROUPS 14 Galois module structures: group representation obstructions.
Wecontinue with the same notation for A , E , K/k , and M . By Proposition 3.2, E admits a model up to isogeny E ∗ defined over a subextension F/k of K/k . We willconsider the following hypotheses:( H ) Gal( K/F ) contains an element τ of order 4.( H ) Gal( K/F ) contains an element τ of order 6.(A ) Gal( K/F ) contains A .(S ) Gal( K/F ) contains S (equivalently, Gal( K/F ) ≃ S ).The main result of this section is the following proposition, which imposes con-ditions on M in terms of Gal( K/F ). Proposition 3.5.
One has:i) If ( H ) holds, then M = Q ( √− or M = Q ( √− .ii) If ( H ) holds, then M = Q ( √− .iii) If (A ) holds, then M splits the quaternion algebra ( − , − Q . iv) If (S ) holds, then M = Q ( √− . The proof is built on some results of [FS14], which we now recall. We note alsothat the proofs of statements i ) and ii ), and of iii ) and iv ) require slightly differenttechniques, so we will treat them separately. First, let us introduce some notationsand make some basic observations. Denote by L/F the minimal extension overwhich Hom( E ∗ L , A L ) ≃ Hom( E ∗ Q , A Q ). Observe that K is contained in L . Lemma 3.6.
One has that
Gal(
L/K ) ⊆ C n , where n is the number of roots ofunity in M .Proof. First observe that
L/F is a Galois extension by the minimality conditiondefining it. Let E be an elliptic curve defined over K such that A K ∼ E . Then L/F may be characterized by the fact that
L/K is the minimal extension overwhich E and E ∗ K become isogenous. Let ψ : E , Q → E ∗ Q be an isogeny. Since G K acts trivially on M = End( E ), the map ξ : G K → M ∗ , ξ ( σ ) := ψ − ❛ σ ψ is a homomorphism. But thenGal( L/K ) ≃ G K / ker( ξ ) ≃ Im( ξ ) ⊆ C n , as desired. (cid:3) Observe that the action of M ≃ Z (End( A Q )) on Hom( E ∗ L , A L ) commutes withthat of G F , since Z (End( A Q )) ⊆ End( A ) by Proposition 2.8. By letting G F actnaturally on the first factor and trivially on M , the tensor productHom( E ∗ L , A L ) ⊗ M,ι M (resp. End( A L ) ⊗ M,ι M ) , taken with respect to any of the two automorphisms ι ∈ { id , c } of M , becomesa M [Gal( L/F )]-module of dimension 2 (resp. 4). Here, c denotes the non-trivialautomorphism of M . Let θ ι ( E ∗ ) (resp. θ ι ( A )) denote the representation affordedby this module.Let π : Gal( L/F ) → Gal(
K/F ) be the natural projection. Note that θ ι ( E ∗ ) is(by the definition of L/F ) a faithful representation of Gal(
L/F ), whereas θ ι ( A ) Recall that M = Q ( √− d ), with d square free, splits ( − , − Q if and only if d To ease notation, for a ∈ M , we will simply write a to denote c a . LLIPTIC k -CURVES AND GENUS 2 SATO–TATE GROUPS 15 is only faithful (by the definition of K/k ) as a representation of Gal(
K/F ). Thefollowing lemma is a restatement of [FS14, Prop. 3.2] and [FS14, Prop. 3.4] in oursetting.
Lemma 3.7.
One has:i)
Tr( θ ι ( A )) = Tr( θ id ( E ∗ )) · Tr( θ c ( E ∗ )) ∈ Q . Thus, θ id ( A ) ≃ θ c ( A ) =: θ ( A ) .ii) For σ ∈ Gal(
L/F ) , let r denote the order of π ( σ ) . Then r is , , , , or ,and Tr( θ ( A )( σ )) = 2 + ζ r + ζ r . Proof of i ) and ii ) of Proposition 3.5. Let r denote the order (4 or 6) of τ . Let˜ τ ∈ Gal(
L/F ) be such that π (˜ τ ) = τ . Let ξ , ω be the eigenvalues of θ ι ( E ∗ )(˜ τ ).Then, i ) and ii ) of Lemma 3.7, give(3.2) ( ξ + ω )( ξ + ω ) = 2 + ζ r + ζ r . Without loss of generality, we may reorder ξ and ω so that (3.2) implies that ξ = ζ r ω . The condition Tr( θ ι ( E ∗ )(˜ τ )) = ω (1 + ζ r ) ∈ M forces ω to belong to the biquadratic extension M ( ζ r ). Therefore, there are onlythe following possibilities for the value of the order t of ω : 1, 2, 3, 4, 6, 8, 12.Suppose that r = 4. It is straightforward to check that: • if t = 1, 2, 4, then ω (1 + ζ r ) ∈ Q ( √− \ Q ; • if t = 8, then ω (1 + ζ r ) ∈ Q ( √− \ Q ; • the values t = 3, 6, 12 are incompatible with ω (1 + ζ r ) belonging to aquadratic imaginary field.Suppose that r = 6. Then one readily checks that • if t = 1, 2, 3, 6, then ω (1 + ζ r ) ∈ Q ( √− \ Q ; • the values t = 4, 8, 12 are incompatible with ω (1 + ζ r ) belonging to aquadratic imaginary field. (cid:3) Remark . One could wonder whether Gal(
K/F ) containing an element τ oforder r = 3 imposes some condition on M . However, the argument of the proof ofProposition 3.5 can not be applied, since ω (1 + ζ r ) make take the rational value ζ (1 + ζ ) = − Lemma 3.9.
One has:i) If (A ) holds and M = Q ( √− , Q ( √− , then Gal(
L/F ) is isomorphic to thebinary tetrahedral group B T .ii) If (S ) holds, then Gal(
L/F ) is isomorphic to either the binary octahedral group B O or to the group GL ( Z / Z ) .Proof. In the course of this proof, let us say that a finite group is 2 -embeddable if it possesses a faithful representation of dimension 2 with coefficients in Q . Incase i ), we have that L/K is at most quadratic by Lemma 3.6. The existence ofthe faithful representation θ ι ( E ∗ ) : Gal( L/F ) → GL ( M ) implies that Gal( L/F ) is2-embeddable (it fact, it implies a bit more: Gal(
L/F ) has a faithful representation The Gap identification numbers of B T , B O , and GL ( Z / Z ) are h , i , h , i , and h , i ,respectively. LLIPTIC k -CURVES AND GENUS 2 SATO–TATE GROUPS 16 of dimension 2 with coefficients in a quadratic imaginary field, but we will notneed that). Since A is not 2-embeddable, we have that L = K . There are twoextensions of A by C : B T and A × C . The lemma then follows from the factthat A × C is not 2-embeddable.As for ii ), we first note that (H ) holds; then M = Q ( √−
1) or M = Q ( √−
2) bystatement i ) of Proposition 3.5. In particular Lemma 3.6 implies that Gal( L/K ) ⊆ C . There are nine groups that are extensions of S by C , none of which is 2-embeddable. Since neither is S , we deduce that L/K is quadratic. Up to isomor-phism, there are four extensions of S by C , only two of which are 2-embeddable:B O and GL ( Z / Z ). (cid:3) Proof of iii ) and iv ) of Proposition 3.5. Assume that we are in case iii ). We maysuppose that M = Q ( √− , Q ( √− θ ι ( E ∗ ) : B T → GL ( M ) . Inspecting the character table of B T , we realize that it has three faithful represen-tations of dimension 2: one has rational trace (call it ̺ ), and the other two havetrace taking values in Q ( √− \ Q on elements of order 3 and 6. Since we haveassumed that M = Q ( √− θ ι ( E ∗ ) ≃ ̺ . Let Q denote the subgroupof B T isomorphic to the quaternion group. It is well-known that the restrictionof ̺ to Q , sometimes called the quaternionic representation of Q , although havingrational trace, is realizable over a field M if and only if M splits the quaternionalgebra ( − , − Q (see [Ser98, Cor. to Prop. 35]).Case iv ) is immediate from Lemma 3.9: one readily checks that the trace ofany faithful representation of dimension 2 of B O or GL ( Z / Z ) takes values in Q ( √− \ Q on elements of order 8. (cid:3) Restriction of scalars: cohomological obstructions.
In this section, let M be an imaginary quadratic field of discriminant D . Assume that(0) D is different from − , −
4, and − A/k is an abelian surface satisfying the following conditions:(1) k contains M ;(2) A K ∼ E , where E/K is an elliptic curve with CM by M ;(3) the field K is the smallest such that End( A K ) = End( A Q ); and(4) the Galois group G = Gal( K/k ) is isomorphic to S .Note that E is then a k -curve completely defined over K (cf. Definition 2.9). Let γ KE ∈ H ( G, M ∗ ) be the cohomology class attached to E as in (2.4). Since K willbe fixed through this section, let us denote γ KE simply by γ E .In § M . For example, since condition (A ) holds, Proposition 3.5 implies that M necessarily splits the quaternion algebra ( − , − Q . The goal of this section is togive some more necessary conditions on M , under the following additional condition(which we assume from now on):(NE) If D is even, then it is either divisible by 8 or by some prime p ≡ E ∗ of E up to isogeny introduced in Remark 3.3. It is any ellipticcurve with CM by the maximal order of M and defined over the field F = k ( j E ∗ ). LLIPTIC k -CURVES AND GENUS 2 SATO–TATE GROUPS 17 We next show that condition (NE) allows for a particular choice of E ∗ that will bekey to our computations.First of all we note that [ F : k ] = 2. Indeed, F/k is a polyquadratic extensionand, since F ⊆ K , we see that [ F : k ] is either 1 or 2. If F = k , then Proposition 3.5implies that M = Q ( √− F : k ] = 2. Let H be the Hilbert class field of M . Thanks to the assumption(NE), by [Gro80, §
11] there exists an elliptic curve ˜
E/H with CM by M that is H -isogenous to its Gal( H/ Q )-conjugates. Since H ⊆ F , we can (and do) take as E ∗ the curve ˜ E F .For this particular choice, E ∗ is a k -curve completely defined over F ; namely, E ∗ is F -isogenous to its Gal( F/k )-conjugate. In particular, the cohomology class γ E ∗ = γ KE ∗ lies in the image of the inflation map H (Gal( F/k ) , M ∗ ) → H ( G, M ∗ ) , and a cocycle c E ∗ representing γ E ∗ is of the form c E ∗ ( σ, τ ) = ( m if σ | F = Id and τ | F = Id , m ∈ M ∗ . Observe that γ E ∗ only depends on the class of m mod ( M ∗ ) ,for replacing c E ∗ by a cohomologous cocycle changes m by an element of ( M ∗ ) .The aim of this section is to prove the following result. Proposition 3.10.
Either m or − m is a square in M . The proof follows from an explicit computation of the decomposition into simplevarieties of the variety R = Res K/k E , the restriction of scalars of E . This isjustified by the observation that A is one of the factors of such decomposition. Lemma 3.11.
The abelian variety A is a simple factor of R . More precisely, R ∼ A × A ′ , for some abelian variety A ′ that does not contain any factor isogenous to A .Proof. By the universal property of the restriction of scalars we have an isomor-phism of Q -vector spacesHom( A, R ) = Hom( A K , E ) ≃ M . (3.4)We claim that End( A ) ≃ M as Q -vector spaces (and, in fact, as Q -algebras).Indeed, the Galois endomorphism type (as described in [FKRS12, § A is F [S ], and then by [FKRS12, Table 8], one deduces that End( A ) has dimension 2over Q . The isomorphism then follows from Proposition 2.8.Therefore, we see that two (and only two) copies of A appear in the decomposi-tion of R into simple varieties and this finishes the proof. (cid:3) In order to further determine the decomposition of R , we will compute thedecomposition of its endomorphism algebra into simple algebras. By [Gro80, § R ) ≃ M c E [ G ] , (3.5) LLIPTIC k -CURVES AND GENUS 2 SATO–TATE GROUPS 18 where M c E [ G ] denotes the twisted group algebra of G by a cocycle c E represent-ing γ E . This is the M -algebra with M -basis the symbols { u σ } σ ∈ G and multiplica-tion given by the rule u σ · u τ = c E ( σ, τ ) u στ . (3.6)In order to relate c E ∗ and c E we need two basic lemmas. Lemma 3.12.
Let
C/K and C ′ /K be elliptic curves with CM by a field M contained in K and different from Q ( √− and Q ( √− . Then there exists anelement β ∈ K such that C ′ is K -isogenous to C β , the K ( √ β ) -twist of C .Proof. Since C and C ′ have CM by M there exists a Q -isogeny λ : C ′ Q → C Q . Thefact that G K acts trivially on M implies that the map G K −→ M ∗ σ λ − ❛ σ λ (3.7)is a homomorphism. Its image is finite, because λ is defined over some finite ex-tension of K , and therefore contained in {± } by our hypothesis on M . If theimage is trivial then λ is defined over K ; otherwise, it is defined over a field of theform L = K ( √ β ) for some β ∈ K ∗ \ ( K ∗ ) . The set Hom(Gal( L /K ) , M ∗ ) = H (Gal( L /K ) , M ∗ ) parametrizes the elliptic curves which are L -isogenous to C ,up to K -isogeny. The curve C β also corresponds to the homomorphism (3.7), andtherefore C ′ is K -isogenous to C β . (cid:3) In the following statement we use the interpretation of H ( G, {± } ) as a groupclassifying (classes of) central extensions of G by {± } . Lemma 3.13.
Let C be a k -curve completely defined over a field K . Supposethat C has CM by a field M contained in k , and let γ C ∈ H (Gal( K /k ) , M ∗ ) be the corresponding cohomology class. Let L = K ( √ β ) for some β ∈ K , and let C β be the L -twist of C . The curve C β is completely defined over K if and onlyif L is Galois over k . Moreover, in that case the cohomology classes γ C , γ C β ∈ H (Gal( K /k , M ∗ )) satisfy the relation γ C β = γ C · γ L , where γ L ∈ H (Gal( K /k ) , {± } ) stands for the cohomology class attached to −→ Gal( L /K ) ≃ {± } −→ Gal( L /k ) −→ Gal( K /k ) −→ . Proof.
This is proved in [GQ14, Lemma 6.1] in the non-CM case. The same argu-ment works in the CM case, with just a small remark: in the course of the argumentof loc. cit. one constructs isogenies ν σ : σ C β → C β and uses the fact that if thecurve is completely defined over K then these ν σ are necessarily defined over K .This is obvious in the non-CM case; in the CM case, if follows from our hypothesisthat M ⊆ k , because then all the endomorphisms of C β are defined over K . (cid:3) Since E and E ∗ are k -curves completely defined over K , the above lemmas implythat E and E ∗ become isogenous over a quadratic extension L/K such that
L/k isGalois, and that γ E = γ E ∗ · γ L , (3.8)where γ L is the cohomology class attached L/K . LLIPTIC k -CURVES AND GENUS 2 SATO–TATE GROUPS 19 Remark . Observe that γ L in principle belongs to H ( G, {± } ); in expressionslike (3.8), we make the slight abuse of notation of using the same symbol to denotethe image of γ L under the natural map H ( G, {± } ) → H ( G, M ∗ )[2] . (3.9)This is justified by the fact that (3.9) is injective. This follows, for instance, fromthe following decomposition, which is analogous to (2.5): H ( G, M ∗ )[2] ≃ H ( G, {± } ) × H ( G, M ∗ / {± } )[2] . Recall that G ≃ S . The cohomology group H (S , {± } ) is known to be iso-morphic to C × C (see, e.g., [Ser84, § L/k ) are also well-known,and we are only interested in the non-symmetric classes, which correspond to B O (the binary octahedral group introduced in § ( Z / Z ). Lemma 3.15.
The cohomology class γ L in (3.8) is non-symmetric. That is, Gal(
L/k ) is isomorphic to B O or GL ( Z / Z ) .Proof. Suppose that γ L is symmetric. It turns out that the symmetric classesin H (S , {± } ) are precisely those that have trivial restriction to A . SinceGal( K/F ) ≃ A , and γ E ∗ has trivial restriction to Gal( K/F ), we see that γ E has trivial restriction to Gal( K/F ). Therefore, E is K -isogenous to a curve definedover F (cf. Remark 2.13). But this is a contradiction with Lemma 3.9, which impliesthat the minimal extension of K over which this can happen is non-trivial. (cid:3) From now on we denote by γ − ∈ H ( G, {± } ) the class corresponding to B O and by γ + the one corresponding to GL ( Z / Z ) (and, as usual c + and c − denotecocycles representing them). So far we have seen that γ E = γ E ∗ · γ ± , where γ E ∗ is given by (3.3) and γ ± ∈ { γ + , γ − } .(3.10)The next step is to compute the center of M c E [ G ]. As a previous calculation, weneed to determine the center of M c ± [ G ]. For this we will use the characterizationof the center of a twisted group algebra M c [ G ] given in [Kar87, p. 321]. Given atwo-cocycle c ∈ Z ( G, M ∗ ), an element g ∈ G is said to be c -regular if c ( g, h ) = c ( h, g ) for all h in the centralizer C G ( g ) of g. (3.11)All conjugates of a c -regular element are also c -regular. Let X denote a set ofrepresentatives of the c -regular conjugacy classes. For x ∈ X let T x denote asystem of representatives of G/C G ( x ). For every x ∈ X , the element k x = X g ∈ T x u g u x u − g (3.12)belongs to the center of M c [ G ]. Moreover, { k x } x ∈ X is an M -basis of the center.Observe that, by making use of (3.6), the element k x can also be expressed as k x = X g ∈ T x u g u x u − g = X g ∈ T x c ( gx, g − ) c ( g, x ) c ( g, g − ) − u gxg − . (3.13) Lemma 3.16.
The center of M c ± [ G ] is isomorphic to M × M [ t ] / ( t ∓ . LLIPTIC k -CURVES AND GENUS 2 SATO–TATE GROUPS 20 Proof.
This is an explicit computation with the cocycle c ± . The computation iselementary, but lengthy. For this reason, we have used the software Magma [BCP97]to carry it out. We reproduce here the details only for c + ; they are very similarfor c − .To begin with, Magma implements routines that allow for the explicit computa-tion the cocycle c + . Alternatively, one can compute it using the exact sequence1 → {± } → GL ( Z / Z ) → S ≃ G → σ ∈ G fix a lift ˜ σ ∈ GL ( Z / Z ); then c + ( σ, τ ) = ˜ σ · ˜ τ · f στ − .Knowing the values c + ( σ, τ ) for all σ, τ ∈ S (here we are identifying G with S ),one can compute the set X : it consists of the conjugacy classes of Id, (1 , , , , ,
2) . This already implies that the center has dimension 3 over M .It remains to compute the structure of the center as an algebra. For this, considerthe element k y with y = (1 , , , c = c + . It turns out that k y = X g ∈ T y c + ( gy, g − ) c + ( g, y ) c + ( g, g − ) − u gyg − (3.14) = u (1 , , , + u (1 , , , + u (1 , , , + u (1 , , , − u (1 , , , − u (1 , , , . (3.15) Using the multiplication formula in the twisted group algebra, namely u σ · u τ = c + ( σ, τ ) u στ , one computes that k y = 3( − u (1 , , + u (1 , , − u (1 , , + u (1 , , − u (1 , , + 2 u Id − u (2 , , + u (1 , , − u (2 , , ) and k y = 18( u (1 , , , + u (1 , , , + u (1 , , , + u (1 , , , − u (1 , , , − u (1 , , , ) . We see that k y − k y = 0 and that the powers of k y do not satisfy any linearrelation of lower degree. Thus the minimal polynomial of k y is t ( t −
18) whichimplies that the center of M c ± [ G ] is isomorphic to M × M [ t ] / ( t − c − gives the minimal polynomial t ( t + 18). (cid:3) Lemma 3.17.
The center of M c E [ G ] is isomorphic to M × M [ t ] / ( t ∓ m ) .Proof. Again we just explicit the calculations for c + ; the case of c − is analogous.By (3.10) we can assume that c E = c E ∗ · c + . Recall that c E ∗ is a symmetric cocycle(cf. formula (3.3)) which lies in the image of the inflation map H (Gal( F/k ) , M ∗ ) → H ( G, M ∗ ) . Since c E ∗ is symmetric, in view of (3.11) an element g ∈ G is c E -regular if and onlyif it is c + -regular. This implies that the center of M c E [ G ] has the same dimensionas the center of M c + [ G ]. We next determine its algebra structure.One of the elements of the center of M c E [ G ] is ˜ k y = X g ∈ T y c E ( gy, g − ) c E ( g, y ) c E ( g, g − ) − u gyg − , where we can take y = (1 , , ,
3) as before. Using formula (3.3) and the fact that g A for g ∈ T y it is easy to check that c E ∗ ( gy, g − ) c E ∗ ( g, y ) c E ∗ ( g, g − ) − = 1 . In https://github.com/xguitart/sato-tate the interested reader can find the Magma scriptthat we used. LLIPTIC k -CURVES AND GENUS 2 SATO–TATE GROUPS 21 Therefore, we have that ˜ k y = X g ∈ T y c + ( gy, g − ) c + ( g, y ) c + ( g, g − ) − u gyg − (3.16) = u (1 , , , + u (1 , , , + u (1 , , , + u (1 , , , − u (1 , , , − u (1 , , , . (3.17) Observe that ˜ k y has the same expression as the element k y found in (3.15) in termsof the basis { u σ } σ ∈ G . We remark that the two elements lie in different algebrasthough: k y lies in M c ± [ G ] and ˜ k y lies in M c E [ G ]. Thus, in order to compute ˜ k y weneed to use now the multiplication of the twisted group algebra M c E [ G ]: u σ · u τ = c E ( σ, τ ) u στ . But observe that the group elements appearing in (3.17) do not belong to A .Therefore, in order to compute ˜ k y only the values c E ( σ, τ ) with σ, τ A areinvolved in the calculation. For σ, τ A we have that c E ( σ, τ ) = m · c + ( σ, τ ).This means that ˜ k y has the same expression in terms of the basis as m · k y ; that is ˜ k y = 3 m ( − u (1 , , + u (1 , , − u (1 , , + u (1 , , − u (1 , , + 2 u Id − u (2 , , + u (1 , , − u (2 , , ) Now the group elements appearing in the above expression of ˜ k y belong to A .Therefore, in the product ˜ k y = ˜ k y · ˜ k y all the products of basis elements are of theform u σ · u τ with σ ∈ A and τ A , so that c E ( σ, τ ) = c + ( σ, τ ). This implies that˜ k y has the same expression in terms of the basis elements as m · k y , namely ˜ k y = 18 m ( u (1 , , , + u (1 , , , + u (1 , , , + u (1 , , , − u (1 , , , − u (1 , , , ) . Therefore, ˜ k y − m ˜ k y = 0 and the minimal polynomial of ˜ k y is t − mt as weaimed to see. (cid:3) From the above lemma we see that M c E [ G ] decomposes into the product ofeither two or three simple algebras, depending on whether ∓ m is a square in M ornot. Before determining the structure of the center, we record the following pieceof information about M c E [ G ], which we will use in § Lemma 3.18.
Each simple factor of M c E [ G ] contains M ( M ) as a subalgebra.Proof. The first step is to prove that M c E [ G ] contains a subalgebra isomorphic toM ( M ). Indeed, let H be the unique normal subgroup of G isomorphic to C × C .The only normal subgroup of S isomorphic to C × C is contained in A ; sinceGal( K/F ) ≃ A we see that H ⊆ Gal(
K/F ). Thus c E ( x, y ) = c ± ( x, y ) for x, y ∈ H because c E ∗ has trivial restriction to Gal( K/F ).Now let s and t be generators of H . The elements u , u s , u t , and u st generatea subalgebra of M c E [ G ] isomorphic to ( a, b ) M with a, b ∈ {± } . Indeed, one cancheck that c ± ( s, s ) ∈ {± } ; c ± ( t, t ) ∈ {± } ; c ± ( s, t ) c ± ( t, s ) = − , (3.18)so that u s = ± , u t = ± u s u t = − u t u s . Therefore, u s and u t generate an algebra which is isomorphic to ( ± , ± M . Aswe have observed before, by Proposition 3.5 iii ) the field M splits the algebra( − , − Q , so that ( ± , ± M ≃ M ( M ). LLIPTIC k -CURVES AND GENUS 2 SATO–TATE GROUPS 22 Let B be a simple factor of M c E [ G ] with projection π : M c E [ G ] → B . Thecomposition M ( M ) ֒ → M c E [ G ] π → B is a homomorphism of M -algebras. Since M ( M ) is simple, it does not have non-trivial ideals and the kernel of the above composition is trivial. Thus the ho-momorphism M ( M ) → B is injective and M ( M ) is a subalgebra of the simplefactor B . (cid:3) Remark . Observe that in the proof of Lemmas 3.17 and 3.18 we have notused that E is the elliptic quotient of A . This will play a role in Proposition 3.20below, but the statements of Lemma 3.17 and Lemma 3.18 are valid for any k -curvecompletely defined over K , with Gal( K/k ) ≃ S and whose cohomology class is asin (3.10). This will be used in § Proposition 3.20.
The center of M c E [ G ] is isomorphic to M × M × M .Proof. From Lemma 3.17 the center of M c E [ G ] is isomorphic to either M × M × M or M × L , with [ L : M ] = 2. Suppose that the center is M × L . This implies that M c E [ G ] ≃ M r ( B ) × M r ( B ) , where B stands for an M -central division algebra, say of Schur index c , and B for an L -central division algebra of Schur index c . Since M c E [ G ] has dimension 24over M we have that r c + 2 r c = 24 . (3.19)Now by Lemma 3.11 one of the simple factors of M c E [ G ] is isomorphic to M ( M ).This forces r = 2 and c = 1, but then (3.19) does not have any solution which isa contradiction. (cid:3) Proof of Proposition 3.10.
The statement is now a direct consequence of Proposi-tion 3.20 and Lemma 3.17. (cid:3)
Some explicit computations.
To illustrate the usefulness of Proposition 3.10we have computed the cohomology class associated to certain curves with CM bya field M of class number 2, in the particular case where k = M . The calculationsthat we next describe have been done using Sage. Let M be one of the imaginary quadratic fields of Table 2. Let j and j be theroots of the Hilbert class polynomial attached to the discriminant D of M . The field Q ( j ) is real quadratic and the Hilbert class field of M is given by H = M · Q ( j ).In particular, since we take k = M , the field F coincides with H .As a first step, we have computed an elliptic curve E / Q ( j ) with j -invariant j (this is easy; one can use for instance the explicit formula of [Sil86, Prop. 1.4]).Then, with a simple search method we have been able to find an element β ∈ Q ( j )such that the curve ( E ) β , the twist of E by β , is F -isogenous to its Gal( F/k )-conjugate. Put E ∗ = ( E ) β , and denote by σ a generator of Gal( F/k ).Using Sage routines we have computed an isogeny µ σ : E ∗ → σ E ∗ explicitly;that is, we have found the rational functions that define µ σ . Then it is easy to The reader can find the scripts in https://github.com/xguitart/sato-tate . Since D satisfies (NE) there exists a curve over F , with CM by M , and which is F -isogenousto its Gal( F/k )-conjugate. However, it is not clear to us that one can always find a model over Q ( j ) of such curve, as we did for the curves of Table 2. LLIPTIC k -CURVES AND GENUS 2 SATO–TATE GROUPS 23 compute its Galois conjugate σ µ σ and the composition σ µ σ ❛ µ σ . Having explicitlythe rational functions giving the isogeny σ µ σ ❛ µ σ allows for the calculation of thekernel polynomial of this isogeny. In each of the entries of Table 2 we have checkedthat such kernel polynomial agrees with the kernel polynomial of the isogeny “mul-tiplication by m ”, where m is the one displayed in the second column. This meansthat σ µ σ ❛ µ σ equals multiplication by ± m (the indeterminacy in the sign comesfrom the fact that the kernel polynomial determines the isogeny up to composingwith − µ σ that we begin with is non-canonical; adifferent choice of µ σ would change m by an element of ( M ∗ ) .As an example, we give the computations for the case k = M = Q ( √− F = Q ( √− , √ E ∗ : y = x + (135 √ − x + 6480 √ − M . There is an isogeny µ σ : E ∗ F → σ E ∗ F withkernel polynomial x + 6 √ − . The Galois conjugate σ µ σ has kernel polynomial x − √ − . One can check that the kernel polynomial of σ µ σ ❛ µ σ equals the kernel polynomialof the multiplication by 2 map on E ∗ .On Table 2, we have computed the value of m mod ( M ∗ ) for some discrimi-nants D of quadratic imaginary fields M of class number 2. Note that for M = Q ( √− Q ( √− Q ( √− m or − m is a square of M ∗ , andthus Proposition 3.10 provides an obstruction to the existence of an abelian surface A/k with Gal(
K/k ) ≃ S and the elliptic quotient E having CM by M . Observe,however, that for D = −
40 and D = −
24, we have that either 2 m or − m is asquare in M , since m = ±
2. Therefore Proposition 3.10 does not yield any ob-struction in these cases. In fact, in § A/k with Gal(
K/k ) ≃ S and the elliptic quotient E having CMby Q ( √− D (discriminant of M ) m mod ( M ∗ ) − ± − ± − ± − ± − ± Table 2.
Values of m mod ( M ∗ ) for certain curves E ∗ with CMby fields of class number 2.3.3. Abelian surfaces over Q . In this section, let A be an abelian surface definedover k satisfying that A Q ∼ E , where E is an elliptic curve over Q with CM, sayby a quadratic imaginary field M . Let K/k be the minimal extension such thatEnd( A K ) ≃ End( A Q ). Denote by M (resp. M ) the finite set of quadraticimaginary fields of class number 1 (resp. of class number 2). We also denote by M , the finite set of imaginary quadratic fields with class group isomorphic to LLIPTIC k -CURVES AND GENUS 2 SATO–TATE GROUPS 24 C × C . On Table 3 below we list the discriminants D of the quadratic imaginaryfields in M , M , and M , , respectively. M − , − , − , − , − , − , − , − , − M − , − , − , − , − , − , − , − , − , − − , − , − , − , − , − , − , − M , − , − , − , − , − , − , − , − , − , − , − , − − , − , − , − , − , − , − , − , − , − , − , − Table 3.
Discriminants of the imaginary quadratic fields withclass group isomorphic to C , C , and C × C . Theorem 3.21.
Let A be an abelian surface defined over a number field k that is Q -isogenous to the square E of an elliptic curve defined over Q with CM by M .If k is either Q or M , the set of possibilities for M provided that Gal(
K/M ) ≃ G is contained in M ( G ) , where the set M ( G ) is as defined on Table 4. Gal(
K/M ) M (Gal( K/M ))C M C M ∪ M C M C { Q ( √− , Q ( √− } ∪ M C { Q ( √− } ∪ M D M ∪ M ∪ M , D M ∪ M D { Q ( √− , Q ( √− } ∪ M ∪ M , D { Q ( √− } ∪ M ∪ M , A M \ { Q ( √− } S { Q ( √− } ∪ M \ { Q ( √− , Q ( √− , Q ( √− , Q ( √− } Table 4.
Possibilities for the field M depending on Gal( K/M ). Proof.
Suppose first that k = M . By Proposition 3.2, E admits a model E ∗ up toisogeny defined over a subextension F/k of K/k , with Gal(
F/k ) ≃ C r for some r ≤ r ( A ), where r ( A ) is as defined in Remark 3.1. By Remark 3.3 we can actuallysuppose that E ∗ is an elliptic curve with CM by the maximal order of M , and that F = k ( j ( E ∗ )). The fact that k = M implies that F is the Hilbert class field of M .Therefore, Gal( F/M ) = Gal(
F/k ) ≃ C r with r ≤ r ( A ) and necessarily M lies in M ∪ M ∪ M , . This proves the rows of Table 4 corresponding to C , C , C ,D , and D .The remaining rows except of the last one follow by combining the above resultwith Proposition 3.5. Indeed, for rows C and D observe that, if M has classnumber 1, then F = M and Gal( K/F ) contains an element of order 4, so that i )of Proposition 3.5 forces M to be Q ( √−
1) or Q ( √− and D are a consequence of ii ) of Proposition 3.5. The row A follows from iii ): Q ( √− − , − Q so it can not appear in the list. LLIPTIC k -CURVES AND GENUS 2 SATO–TATE GROUPS 25 Finally, for the row S , besides from the above considerations, one needs toinvoke Proposition 3.10 and Table 2. Indeed, first note that if the class numberis 1, then M can only be Q ( √−
2) because of statement iv ) of Proposition 3.5;second, Q ( √−
15) can not appear, because it does not split ( − , − Q ; third, thequadratic imaginary fields of class number 2 and discriminants D = − D = − D = −
115 cannot occur because the respective values of ± m (as listed onTable 2) are not squares in M .This finishes the case where k = M . The case k = Q now follows by applyingthe previous case to the base change A M . (cid:3) We note that Table 4 is to be read as follows: if M does not belong to M ( G ),then there does not exists any A/k such that Gal(
K/M ) ≃ G and its absolutelysimple factor has CM by M . We do not claim, however, that for any pair ( G, M )with M ∈ M ( G ) there does exist such an A . In other words, it might be that someof the pairs of ( G, M ) in Table 4 cannot be realized by any abelian surface.In § G, M ). For example, one can performa search over equations of genus 2 curves and try to identify the endomorphismalgebra of its Jacobian and the minimum field of definition of its endomorphisms;or one can look for equations over families of genus 2 curves parametrizing thosewith a specified group of automorphisms. A slightly less explicit method (in thesense that one does not get the equation of a genus 2 curve out of it) is to find A as a simple factor of the restriction of scalars of a suitable elliptic curve with CMby M . We illustrate this in the following remark, which sumarizes a result of T.Nakamura in [Nak04]. Remark . The pair (
G, M ) = (C × C , Q ( √− M . Indeed, let K be the Hilbert class field of M , so thatGal( K/M ) ≃ C × C , and let E be an elliptic curve with CM by M and with theproperty that E is defined over K and it is K -isogenous to all of its Gal( K/ Q )-conjugates. Such elliptic curves exist; in fact, in [Nak04, p. 190] it is shown thatthere are 8 of them, up to K -isogeny. Let R = Res K/M E denote the restrictionof scalars, which is an abelian variety over M of dimension 4. Nakamura showsthat End( R ) is isomorphic to the quaternion algebra ( a, b ) M , where the pair ( a, b )is one of the entries in the table of [Nak04, p. 190]. It turns out that, for allthe entries in that table, there is an isomorphism ( a, b ) M ≃ M ( M ), and therefore R ∼ A for some A/M such that A K ∼ E . It follows from the theory of complexmultiplication that K is the smallest field of definition of the endomorphisms of A .A similar approach can be used to realize the pair (S , Q ( √− § An abelian surface with
Gal(
K/k ) ≃ S and M = Q ( √− . In this sectionwe provide an example of abelian surface
A/k that satisfies conditions (0), (1), (2),(3), and (4) described at the beginning of § K , k , and M , and of the curve E/K . Then weconstruct A as a simple factor of the restriction of scalars of E , a step in which wewill take advantage of the explicit calculations performed in § LLIPTIC k -CURVES AND GENUS 2 SATO–TATE GROUPS 26 We begin by describing the number fields involved in the construction. Let K bethe number field with defining polynomial x − x +5 x − x +2. The Galois closure K ′ of K has Galois group isomorphic to S and contains Q ( √
5) as a subfield.Consider also the imaginary quadratic field M = Q ( √− k = M and K = K ′ · k . The splitting field of the Hilbert class polynomialattached to D = −
40 is Q ( √ F = M · Q ( √
5) is the Hilbert class fieldof M . Thus the diagram of fields is the following: K F = M · Q ( √ k = M = Q ( √− Q ( √ PPPPPPPPPPPP Q PPPPPPPPPPPPPP with Gal( K/k ) ≃ S and Gal( K/F ) ≃ A . For future reference we also put K = K · k (note that K is of degree 4 over k and its Galois closure is K ).The discriminant of M satisfies condition (NE) of § E ∗ /F which is F isogenous to its Gal( F/k )-conjugate. Thus E ∗ is a k -curvecompletely defined over F and one can attach to it a cohomology class γ FE ∗ , whichlies in H (Gal( F/k ) , M ∗ ) and it is non-trivial since E ∗ does not admit a modelup to isogeny over k . In fact, it will be more convenient for us to regard E ∗ as a k -curve completely defined over K . In particular, we will be concerned with thecohomology class γ KE ∗ ∈ H (Gal( K/k ) , M ∗ ), which is nothing more than the imageof γ FE ∗ under the inflation map H (Gal( F/k ) , M ∗ ) → H (Gal( K/k ) , M ∗ ) . Throughout this section, let us simply write γ E ∗ = γ KE ∗ . Therefore, as in (3.3), thecocycle c E ∗ representing γ E ∗ is of the form c E ∗ ( σ, τ ) = ( m if σ | F = Id and τ | F = Id , m ∈ M ∗ \ ( M ∗ ) . The cohomology class γ E ∗ is determined by the ele-ment m of (3.20) (or rather by its class modulo ( M ∗ ) ). This is precisely what wecomputed in the third entry of Table 2; we record the result for future reference. Lemma 3.23.
The element m of (3.20) equals either or − , up to multiplicationby an element of ( M ∗ ) . The curve E ∗ is the starting point of our the construction of A . The nextstep is to consider a certain twist of E ∗ , associated to the solution of a suitableGalois embedding problem of K/k . Recall that the group H (S , {± } ), which isisomorphic to C × C , classifies central extensions of S by {± } . We are interested This is the field [LMFDB, Global Number Field 4.0.5780.1], and it is contained in the degree8 field [LMFDB, Global Number Field 8.0.835210000.1]
LLIPTIC k -CURVES AND GENUS 2 SATO–TATE GROUPS 27 in the two cohomology classes γ + and γ − , introduced in § ( Z / Z ) and B O .Given a cohomology class γ ∈ H (Gal( K/k ) , {± } ) a quadratic extension L/K is said to be a solution to the embedding problem defined by γ if the extension1 −→ {± } ≃ Gal(
L/K ) −→ Gal(
L/k ) res −→ Gal(
K/k ) −→ γ . From now on we regard γ + and γ − as elements of H (Gal( K/k ) , {± } ) by means of an identification Gal( K/k ) ≃ S . In this way, γ + and γ − define two embedding problems of Gal( K/k ). Lemma 3.24.
There exist solutions to the embedding problems associated to γ + and γ − .Proof. Let Q K be the quadratic form x Tr K /k ( x ) and let w ( Q K ) denote itsHasse–Witt invariant. We denote by d K the discriminant of K /k . By [Que95,Thm. 3.8] the embedding problems corresponding to B O and to GL ( Z / Z ) aresolvable if and only if the following elements in the Brauer group of k are trivial: w ( Q K ) ⊗ (2 , d K ) and w ( Q K ) ⊗ ( − , d K ) . (3.21)Since K = K · k the above classes can be regarded as w ( Q K ) k ⊗ (2 , d K ) k and w ( Q K ) k ⊗ ( − , d K ) k , (3.22)where the index k stands for the image under the natural map Br( Q ) → Br( k ).One checks that w ( Q K ) is ramified only at 2 and ∞ , and therefore w ( Q K ) k istrivial (since k is imaginary and 2 ramifies in k ). Also, the algebras ( ± , d K ) aresplit by k , so that ( ± , d K ) k are trivial. (cid:3) Let γ ± ∈ H (Gal( K/k ) , {± } ) be the cohomology class defined as follows: γ ± = ( γ + if m = 2 mod ( M ∗ ) γ − if m = − M ∗ ) . (3.23)Let L = K ( √ β ) be a solution to the embedding problem associated to γ ± anddefine E = ( E ∗ ) β , the quadratic twist of E ∗ by β . Since L/k is Galois, Lemma 3.13implies that E is a k -curve completely defined over K with cohomology class γ E = γ E ∗ · γ ± . (3.24)Let R = Res K/k E . In view of Remark 3.19, the statements of Lemma 3.17 andLemma 3.18 are valid in the present context, and they allow for the computationof End( R ). Lemma 3.25.
The center of
End( R ) is isomorphic to M × M × M ,Proof. By Lemma 3.17 the center of M c E [Gal( K/k )] is isomorphic to M × M [ t ] / ( t ∓ m ) . (3.25)By Lemma 3.23 we have that m = 2 or m = −
2, and thanks to our choice of γ ± in(3.23) we see that (3.25) becomes M × M [ t ] / ( t − ≃ M × M × M. (cid:3) LLIPTIC k -CURVES AND GENUS 2 SATO–TATE GROUPS 28 Proposition 3.26.
One has:
End( R ) ≃ M ( M ) × M ( M ) × B, (3.26) where B is an M -central simple algebra of M -dimension .Proof. By the above lemma M c E [Gal( K/k )] is the product of three M -central sim-ple algebras. Denote by r , r , and r the M -dimension of each simple factor. Sincethe dimension of M c E [Gal( K/k )] is 24 we have that r + r + r = 24 . The only solution (up to permutation) is r = 2, r = 2 and r = 4. That is tosay, two factors have dimension 4 and the other factor has dimension 16. Sinceeach simple factor contains M ( M ) by Lemma 3.18, we see that the algebras ofdimension 4 must be isomorphic to M ( M ), and this proves the proposition. (cid:3) The decomposition of End( R ) provided by Proposition 3.26 implies an isotypicaldecomposition of R of the form R ∼ A × A × A r , (3.27)where the A i are simple abelian varieties, A and A correspond to the factors ofthe form M ( M ), and r is equal to either 1, 2 or 4. Let A be either A or A . Wenext show that A satisfies the following properties: • End( A ) ≃ M ; • A K ∼ E ; and • K is the smallest extension of k satisfying that End( A K ) = End( A Q ).The first statement above is clear in light of (3.26). We prove the other two in thefollowing lemmas; this will finish the proof that A is an abelian surface satisfyingconditions (0)–(4) stated at the beginning of § Lemma 3.27. A is an abelian surface; that is, A K ∼ E .Proof. By the universal property of the restriction of scalars functor we have anisomorphism of vector spacesHom(
A, R ) ≃ Hom( A K , E );from (3.27) and (3.26) we see that Hom( A, R ) ≃ M and therefore Hom( A K , E ) ≃ M , which implies that A K ∼ E . (cid:3) Lemma 3.28.
There is no field N with k ( N ( K such that End( A N ) =End( A Q ) .Proof. Suppose that such N exists. Without loss of generality we can assume thatit is the smallest field satisfying this property. This implies, in particular, that N/k is a Galois extension. Then A N ∼ C , for some elliptic curve C over N .Since A K ∼ E this implies that C K ∼ E . Therefore, the cohomology class γ E is trivial when restricted to Gal( K/N ). But Gal(
K/N ) is a normal subgroup ofGal(
K/k ) ≃ S , and all normal subgroups of S are contained in A ; thus Gal( K/N )is a normal subgroup of Gal(
K/F ) ≃ A . Since γ E = γ E ∗ · γ ± and the restrictionof γ E ∗ to Gal( K/F ) is trivial, this implies that γ ± restricted Gal( K/N ) is trivial.This is a contradiction, because it turns out that neither γ + nor γ − become trivialwhen restricted to any normal subgroup of A . Indeed, let H = h s, t i denote theunique normal subgroup of Gal( K/k ) isomorphic to C × C . By looking at the LLIPTIC k -CURVES AND GENUS 2 SATO–TATE GROUPS 29 diagram of subgroups of S we see that Gal( K/N ) must contain H . But we havealready seen in (3.18) that γ ± restricted to H is non-trivial. (cid:3) Two applications to Sato–Tate groups
In this section, we give two applications of the results obtained so far to thetheory of Sato–Tate groups of abelian surfaces over number fields. The Sato–Tategroup of an abelian surface A defined over the number field k , which we will denoteby ST( A ), is a compact real Lie subgroup of USp(4) that conjecturally governsthe distribution of the Frobenius elements in the image of ℓ -adic representationattached to A . We refer to [FKRS12, §
2] for the precise construction of the Sato–Tate group of an abelian variety defined over a number field. The reader is referredto the original source [Ser12, Chap. 8] for a construction of the Sato–Tate groupin a more general context.The main result of [FKRS12] establishes the existence of 52 possibilities for theSato–Tate group of an abelian surface defined over a number field, all of whichoccur for some choice of the number field and the abelian surface (see [FKRS12,Thm. 4.3]). This result is complemented by establishing a one-to-one correspon-dence between Sato–Tate groups of abelian surfaces and their
Galois endomorphismtypes , an algebraic structure which encaptures both the Galois action on the ring ofendomorphisms of the abelian surface and the structure of this ring as an R -algebra(see [FKRS12, Def. 1.3] for a precise definition). Let N ST , ( k ) denote the numberof subgroups of USp(4) up to conjugacy that arise as Sato–Tate groups of abeliansurfaces defined over the number field k .We will make use of the notations for Sato–Tate groups and Galois endomorphismtypes introduced in [FKRS12, §
4] from now on. On Table 5 we show the dictionarybetween Sato–Tate groups and Galois endomorphism types. The groups decoratedwith a ⋆ are those that arise over Q . Note that we thus have N ST , ( Q ) = 34.There are 6 possibilities for the connected component of the identity of the Sato–Tate group of an abelian surface A defined over k . As customary, we will denote itby ST( A ) . The dictionary between Sato–Tate groups and Galois endomorphismtypes establishes that Sato–Tate groups with identity component isomorphic tothe unitary group U(1) of degree 1 correspond to absolute Galois endomorphismtype F , that is, to abelian surfaces that are Q -isogenous to the square of an ellipticcurve with CM by a quadratic imaginary field M . A Galois endomorphism type ofabsolute type F consists of the data(4.1) F [Gal( K/k ) , Gal(
K/kM ) , B ] , where K/k is the minimal extension over which all the endomorphisms of A aredefined, and B is H or M ( R ) depending on whether there exists a subextension K /k of K/k such that End( A K ) ⊗ R is isomorphic to H or not. We removeGal( K/kM ) from the notation in case that it coincides with Gal(
K/k ), and similarlywe avoid writing B when the other data is enough to uniquely determine the Galoisendomorphism type.4.1. A finiteness result.
The first application to the theory of Sato–Tate groupsis now an immediate corollary of Theorem 3.21.
Theorem 4.1.
Among the possibilities for the Sato–Tate group of an abeliansurface A defined over Q , the with identity component isomorphic to U(1) only
LLIPTIC k -CURVES AND GENUS 2 SATO–TATE GROUPS 30 G Galois type G Galois type C F [C ] D , ⋆ F [D , D ] C F [C ] D , ⋆ F [D , D , M ( R )] C F [C ] D , ⋆ F [D , C ] C F [C ] D , ⋆ F [D , C ] C F [C ] D , ⋆ F [D , C ] D F [D ] O ⋆ F [S , A ] D F [D ] E ⋆ E [C ] D F [D ] E ⋆ E [C , C ] D F [D ] E ⋆ E [C ] T F [A ] E ⋆ E [C ] O F [S ] E ⋆ E [C ] J ( C ) F [C , C , H ] J ( E ) ⋆ E [C , R × R ] J ( C ) ⋆ F [D , C , H ] J ( E ) ⋆ E [D ] J ( C ) F [C , C , H ] J ( E ) ⋆ E [D ] J ( C ) ⋆ F [C × C , C ] J ( E ) ⋆ E [D ] J ( C ) ⋆ F [C × C , C ] J ( E ) ⋆ E [D ] J ( D ) ⋆ F [D × C , D ] F D [C ] J ( D ) ⋆ F [D , D , H ] F a D [C , R × C ] J ( D ) ⋆ F [D × C , D ] F ab D [C , R × R ] J ( D ) ⋆ F [D × C , D ] F ac⋆ D [C ] J ( T ) ⋆ F [A × C , A ] F a,b⋆ D [D ] J ( O ) ⋆ F [S × C , S ] G , C [C ] C , ⋆ F [C , C , M ( R )] N ( G , ) ⋆ C [C ] C , F [C , C ] G , ⋆ B [C ] C , ⋆ F [C , C , M ( R )] N ( G , ) ⋆ B [C ] D , ⋆ F [D , C , M ( R )] USp(4) ⋆ A [C ] Table 5.
Sato–Tate groups and Galois endomorphism types ofabelian surfaces occur among the set (of cardinality at most ) of Q -isogeny classes of abeliansurfaces over Q that are Q -isogenous to the square of an elliptic curve definedover Q with CM by M in M ∪ M ∪ M , . More precisely, for such an abeliansurface A , the set of possibilities for M provided that ST( A M ) / ST( A M ) ≃ G iscontained in M ( G ) .Proof. We have already mentioned that an abelian surface whose Sato–Tate grouphas identity component isomorphic to U(1) has absolute Galois endomorphismtype F . Observe thatGal( K/M ) ≃ ST( A M ) / ST( A M ) ≃ ST( A ) ns / ST( A ) ns, . The first isomorphism is [FKRS12, Prop. 2.17]. For the second, as well as forthe definition of ST( A ) ns , see the paragraph following [FKRS12, Rem. 4.10]. Oneconcludes by readily checking (for example from [FKRS12, Table 2]) that the Sato–Tate groups G on the first column and the i th row of Table 6 are precisely thosefor which G ns /G ns, is isomorphic to the finite group appearing in the first columnand the i th row of Table 4. (cid:3) A number field for all genus 2 Sato-Tate groups.
The second applicationand the main result of the present work is the following.
LLIPTIC k -CURVES AND GENUS 2 SATO–TATE GROUPS 31 Sato–Tate groups M (Gal( K/M )) C , M J ( C ) , D , M ∪ M C , , D , M J ( C ) , D , { Q ( √− , Q ( √− } ∪ M J ( C ) , D , { Q ( √− } ∪ M J ( D ) , D , M ∪ M ∪ M , J ( D ) , D , M ∪ M J ( D ) { Q ( √− , Q ( √− } ∪ M ∪ M , J ( D ) { Q ( √− } ∪ M ∪ M , J ( T ) , O M \ { Q ( √− } J ( O ) { Q ( √− } ∪ M \ { Q ( √− , Q ( √− , Q ( √− , Q ( √− } Table 6.
Possibilities for the field M depending on the Sato–Tate group. Theorem 4.2.
Set k := Q ( √− , √− , √− , √− , √ · , √− . Then, N ST , ( k ) = 52 . That is, there exist abelian surfaces defined over k realizing each of the possible Sato–Tate groups of abelian surfaces defined overnumber fields. Hereafter, we will be concerned with the proof of the previous theorem. Wewill assemble our constructions of abelian surfaces into six main families. Beforeproceeding to the proof, we make the following remark.
Remark . Observe that in § F [S ] (equiv. Sato–Tate group O ) and M = Q ( √− F [S × C , S ] or F [S , A ] (equiv. Sato–Tate groups J ( O ) or O )that we are aware of have M = Q ( √− k tocontain Q ( √−
40) and not to contain Q ( √− K contains either Q ( i ) or Q ( √− k , we require k not to contain the two latter fields. The restriction of scalars construction.
As mentioned in the above remark,the abelian surface A over k = Q ( √−
40) defined in § O .Since K ∩ k = k , it follows from [FKRS12, Prop. 2.17] that the Sato–Tate groupof the base change A k remains the same. Base change constructions.
We consider the set of curves Σ Q of [FKRS12,Table 11] which are defined over k = Q . Note that Σ Q has cardinality 34. Wehave used the software Sage to check that for each C ∈ Σ Q the number field K In https://github.com/xguitart/sato-tate the interested reader can find the Sage scriptthat we used. LLIPTIC k -CURVES AND GENUS 2 SATO–TATE GROUPS 32 attached to Jac( C ) satisfies that K ∩ k = Q . It then follows that ST(Jac( C )) ≃ ST(Jac( C ) k ). Twisting constructions.
We will construct examples of abelian surfaces real-izing the Sato–Tate groups C n for n ∈ { , , } , D n for n ∈ { , } , and T by usingthe twisting procedure of [MRS07] or [Mil72, § E defined over k withCM by an order O in a quadratic imaginary field M ⊆ k , a finite Galois extension L/k , and an Artin representation ̺ : Gal( L/k ) → GL ( O ) with coefficients in O ,there is an abelian surface A := E ⊗ ̺ defined over k such that A L ∼ E L and thereis an isomorphism of G k -modules(4.2) End( A Q ) ≃ ̺ ⊗ ̺ ∗ ⊗ End( E Q ) , where ̺ ∗ denotes the contragredient representation of ̺ ([MRS07, Prop. 1.6. i)]).For example, we may take E to be an elliptic curve with CM by M := Q ( √− ⊆ k and defined over k . It then follows from (4.2) that the minimal extension K/k over which all the endomorphisms of A are defined is the extension cut out by ̺ ⊗ ̺ ∗ . For each H ∈ { C n for n ∈ { , , } , D n for n ∈ { , } , and A } take a Galois extension L/k such that Gal( L/k ) be, respectively, isomorphic to˜ H ∈ { C n for n ∈ { , , } , D n for n ∈ { , } , and B T } . Take a faithful rational Artin representation ̺ of degree 2 of ˜ H . By inspection ofthe character table of H one can compute the trace of ̺ ⊗ ̺ ∗ and see that in all thecases the kernel N of ρ ⊗ ̺ ∗ has order 2, and that the quotient ˜ H/N is isomorphicto H ; thus, the field cut by ρ ⊗ ̺ ∗ has Galois group isomorphic to H . It followsfrom the description given in (4.1) that the Galois endomorphism type of A is F [ H ].According to Table 5, we have realized the desired Sato–Tate groups. Remark . One may try to make similar constructions to realize the groups C , D , C , D , and O over k . However, in these cases, the corresponding group ˜ H does not possess a faithful rational degree 2 representation, but rather one withcoefficients in the ring of integers of the fields Q ( i ) or Q ( √− E to have CM by them, and thus k to contain them. This is a possibility that we have excluded in Remark 4.3. Cardona and Quer’s constructions.
We will construct curves whose Jaco-bians have Sato–Tate groups C , D , C , D , and J ( C ), respectively. For thiswe will recall results of Cardona and Quer parametrizing Q -isomorphism classesof genus 2 curves with prescribed automorphism groups. If y = f ( x ), with f ( x ) ∈ k [ x ], is a defining equation for a genus 2 curve C defined over k , thereis an injective group homomorphismAut( C Q ) ֒ → GL ( Q )that sends an automorphism( x, y ) (cid:18) mx + npx + q , mq − np ( px + q ) y (cid:19) , where m, n, p, q ∈ Q , Note that, since k contains neither Q ( i ), nor Q ( √− Q ( √− k in such a way that the Sato–Tate group is preserved. LLIPTIC k -CURVES AND GENUS 2 SATO–TATE GROUPS 33 to the matrix (cid:18) m np q (cid:19) . From now on, we will use matrix notation to write auto-morphisms of genus 2 curves.The next proposition encompasses weakened versions of [CQ07, Prop. 4.3],[CQ07, Prop. 4.9], and [Car01, Thm. 7.4.1] which are enough for our purposes. Proposition 4.5.
Let u, v ∈ Q ∗ and s, z ∈ Q .i) If − z u = s uv , then the curve given by the affine equation (4.3) C D : y = (1 + 2 uz ) x − suvx + v (3 − uz ) x ++ v (3 − uz ) x + 8 suv x + v (1 + 2 uz ) has automorphism group isomorphic to D . It is generated by the matrices (4.4) U = (cid:18) α ββ/v − α (cid:19) , V = (cid:18) −√ v / √ v (cid:19) , where (4.5) α = r − z √ u , β = r v (1 + z √ u )2 . ii) If u − z = 3 s v , the curve given by the affine equation (4.6) C D : y = 27( u + 2 z ) x − svx + 27 v ( u − z ) x + 360 sv x +9 v ( u + 10 z ) x − sv x + v ( u − z ) has automorphism group isomorphic to D . It is generated by the matrices (4.7) U = 1 √ u (cid:18) α β β/v − α (cid:19) , V = 12 (cid:18) √ v − / √ v (cid:19) , where α, β ∈ Q are such that (4.8) α − u α − z , α + 3 β v = u. iii) If u − z = 3 s v , the curve given by the affine equation (4.9) C : y = 27 zx − svx − vzx ++180 sv x + 45 v zx − sv x − v z has automorphism group isomorphic to . It is generated by the matrices (4.10) U = 1 √ u (cid:18) α β β/v − α (cid:19) , V = √− (cid:18) −√ v/ / √ v (cid:19) , where α, β ∈ Q are such that (4.11) α − u α − z , α + 3 β v = u. Remark . We wish to warn the reader of a minor misprint in the work of Cardonaand Quer. In [CQ07, Prop. 3.5] the lower left entry of matrix U is missing a factorof 3 (compare with (4.7) above); we have introduced a similar correction to thesecond equation of both (4.8) and (4.11). The group 2D is a certain double cover of D . Its GAP identification number is h , i . LLIPTIC k -CURVES AND GENUS 2 SATO–TATE GROUPS 34 Remark . It is well known and easy to show that an automorphism group struc-ture of the type D , D or 2D on a genus 2 curve induces a decomposition up to Q -isogeny of its Jacobian as the square of an elliptic curve E . In the 2D case, E has automatically CM by Q ( √− u ∈ Q ∗ for which E has CM in the D and D cases. For example,in the former case, for u = 81 / E has CM by Q ( √− u = 4 / E has CM by Q ( √− Lemma 4.8.
Let C denote either C D , or C D , or C . Suppose that z = 0 andthat Jac( C ) is Q -isogenous to the square of an elliptic curve with CM by M . Thenthe minimal extension of Q over which all the endomorphisms of Jac( C ) are definedis K = M ( α, β, √ u, √ v ) .Proof. By Remark 4.7, K is the composition K ′ of M and the minimal field overwhich all the automorphisms of C are defined. The case D is then immediate. Forthe remaining two cases, we need to check that K ′ = M ( α/ √ u, β/ √ u, √ v ) agreeswith the expression for K given in the statement. But this follows from the factthat α = (cid:18) α − u (cid:19) − z ∈ K ′ , since α = u ( α/ √ u ) ∈ K ′ . (cid:3) On Table 7, C denotes one of the curves C D , C D , or C for the choice ofparameters s = 1, u and z as specified on the second and third columns, and v asdetermined by the constraints of Proposition 4.5. The fourth and fifth columns arecomputed using Lemma 4.8. Together with Remark 4.7, they imply all but the lastrow of the last column. C u z K ∩ k Gal( Kk /k ) ST( C k ) C D /
320 1 Q ( √−
40) D D C D / − / Q ( √−
40) C C C D /
17 1 Q ( √−
51) D D C D / − / Q ( √−
51) C C C
19 19 / Q ( √−
57) C J ( C ) Table 7.
A few examples from Cardona’s parametrizations.Suppose that C is as specified in the last row of Table 7. To verify the bot-tom right entry, it suffices to show that End(Jac( C K )) ⊗ R ≃ H , where K = Q ( √− , α ). This can be easily deduced from the fact thatAut( C K ) ≃ Dic , where Dic is the Dicyclic group of order 12 (with GAP identification number h , i ). This is a group whose faithful representation of degree 2 has Frobenius–Schur index − ( R ). Sutherland’s additional examples.
Besides the examples showed on [FKRS12,Table 11], the wide search performed in [FKRS12], yielded examples that will beuseful to realize J ( C ) and C , over k . We thank Drew Sutherland for pointingout to us the two curves in this paragraph. LLIPTIC k -CURVES AND GENUS 2 SATO–TATE GROUPS 35 Consider the curve C : y = 3 x + 16 x − x − x − x + 3 . We first show that the minimal extension of Q over which all of the endomorphismsof Jac( C ) are defined is K = Q ( √− , √− K isminimal with the property Aut( C K ) ≃ GL ( Z / Z ) . Indeed, any genus 2 curve C such that Aut( C Q ) ≃ GL ( Z / Z ) is Q -isomorphic to y = x − x . Since Gal( Kk /k ) ≃ C , in order to show that ST(Jac( C k )) = J ( C ),all we need to check is that End(Jac( C Q ( √− ) ⊗ R ≃ H . But this follows easilyfrom Aut( C Q ( √− ) ≃ SL ( Z / Z ) . Indeed, SL ( Z / Z ) contains the group of quaternions Q , whose rational faithfuldegree 2 representation has Frobenius–Schur index −
1. Thus SL ( Z / Z ) can notbe embedded in GL ( R ).Consider now the curve C : y + ( x + x + x + 1) y = − x − x − x − x This is the genus 2 curve [LMFDB, Genus 2 Curve 40000.e.200000.1]. To showthat the minimal extension of Q over which all of the endomorphisms of Jac( C ) aredefined is K = Q [ x ] / ( x − x + 4 x − x + 16), we again check that K is minimalwith the property that Aut( C K ) ≃ GL ( Z / Z ) . Since Gal( Kk /k ) ≃ C , in order to show that ST(Jac( C k )) = C , , all weneed to check is that End(Jac( C K ) ⊗ R ≃ H , where K is the index 2 subfield of K given by Q [ x ] / ( x + 2 x + 4 x + 8 x + 16). But this follows easily fromAut( C K ) ≃ SL ( Z / Z ) , and the same argument used for the previous curve. Product constructions.
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LLIPTIC k -CURVES AND GENUS 2 SATO–TATE GROUPS 37 Departament de Matem`atiques, Universitat Polit`ecnica de Catalunya, Edifici Omega,C/Jordi Girona 1–3, 08034 Barcelona, Catalonia
E-mail address : [email protected] URL : https://mat-web.upc.edu/people/francesc.fite/ Departament d’ `Algebra i Geometria, Universitat de Barcelona, Gran via de lesCorts Catalanes, 585, 08007 Barcelona, Catalonia
E-mail address : [email protected] URL ::