Given an elliptic curve $E/k$ and a Galois extension $k'/k$, we construct an
exact functor from torsion-free modules over the endomorphism ring ${\rm
End}(E_{k'})$ with a semilinear ${\rm Gal}(k'/k)$ action to abelian varieties
over $k$ that are $k'$-isogenous to a power of $E$. As an application, we show
that every elliptic curve with complex multiplication geometrically is
isogenous over the ground field to one with complex multiplication by a maximal
order.Comment: 6 pages, added reference

Vogt, Isabel

English

arXiv

Isabel Vogt

Journal de Théorie des Nombres de Bordeaux

Isabel VOGTAbelian varieties isogenous to a power of an elliptic curve over a GaloisextensionTome 31, no 1 (2019), p. 205-213.<http://jtnb.cedram.org/item?id=JTNB_2019__31_1_205_0>© Société Arithmétique de Bordeaux, 2019, tous droits réservés.L’accès aux articles de la revue « Journal de Théorie des Nom-bres de Bordeaux » (http://jtnb.cedram.org/), implique l’accordavec les conditions générales d’utilisation (http://jtnb.cedram.org/legal/). Toute reproduction en tout ou partie de cet article sousquelque forme que ce soit pour tout usage autre que l’utilisation àfin strictement personnelle du copiste est constitutive d’une infrac-tion pénale. Toute copie ou impression de ce fichier doit contenir laprésente mention de copyright.cedramArticle mis en ligne dans le cadre duCentre de diffusion des revues académiques de mathématiqueshttp://www.cedram.org/Journal de Théorie des Nombresde Bordeaux 31 (2019), 205–213Abelian varieties isogenous to a power of anelliptic curve over a Galois extensionpar Isabel VOGTRésumé. Soient E/k une courbe elliptique et k′/k une extension de Galois.On construit un foncteur exact de la catégorie des modules sans torsion surl’anneau des endomorphismes EndEk′ munis d’une action semi-linéaire deGal(k′/k) vers la catégorie des variétés algébriques sur k qui sont k′-isogènesà une puissance de E. Comme application, on donne une preuve simple dufait que toute courbe elliptique sur k qui est géométriquement à multiplicationcomplexe, est isogène sur k à une courbe elliptique à multiplication complexepar un ordre maximal.Abstract. Given an elliptic curve E/k and a Galois extension k′/k, weconstruct an exact functor from torsion-free modules over the endomorphismring EndEk′ with a semilinear Gal(k′/k) action to abelian varieties over k thatare k′-isogenous to a power of E. As an application, we give a simple proof thatevery elliptic curve with complex multiplication geometrically is isogenousover the ground field to one with complex multiplication by a maximal order.1. IntroductionLet E be an elliptic curve over a field k. As in [4], the theory of abelianvarieties isogenous over k to a power of E is related to the theory of finitelypresented torsion-free modules over the endomorphism ring Rk := Endk E.To recall briefly, there is a functorHomRk( · , E) :{finitely presentedleft Rk-modules}opp→{commutative properk-group schemes},such that for M a finitely presented Rk-module and C a k-scheme, we haveHomk(C,HomRk(M,E)) = HomRk(M,Homk(C,E)).Restricting to torsion-free modules, we obtain a functorHomRk( · , E) :{fin. pres. tors. freeleft Rk-modules}opp→{abelian k-varietiesisogenous to Er, r ∈ Z}.Manuscrit reçu le 29 avril 2018, accepté le 19 octobre 2018.2010 Mathematics Subject Classification. 14K02, 11G10.Mots-clefs. abelian varieties, complex multiplication, isogenies.This research was partially supported by the National Science Foundation Graduate ResearchFellowship Program under Grant DMS-1122374 as well as Grant DMS-1601946.206 Isabel VogtThis functor is fully faithful, but not in general an equivalence of categories.For example, if chat k = 0 and E does not have complex multiplication,then the image of this functor consists entirely of the powers of E; yet it ispossible for E to be k-isogenous to a non-isomorphic curve.In fact, this functor is never surjective if E/k acquires complex mul-tiplication (CM) over a separable quadratic extension k′/k: E is alwaysk-isogenous to its k′/k quadratic twist E′/k, as we now show. Suppose thatEk′ has CM by the order O. Let α ∈ O be a purely imaginary element ofO, so that the nontrivial element σ ∈ Gal(k′/k) acts as σα = −α. Multi-plication by α composed with the isomorphism Ek′'−→ E′k′ is Galois stable,and so descends to an isogeny E → E′ over k.In this note we describe a generalization of the functor HomRk( · , E),which in particular addresses the case when Endk E 6= Endk̄ E. We willshow that the essential image of this functor always contains all of thequadratic twists of E.More generally, one may also consider abelian varieties, as in the caseof nontrivial twists, that are isogenous to a power of an elliptic curve onlyafter passing to a Galois extension of the ground field. When restricted totorsion-free modules, the image of the functor we construct will lie in thecategory of abelian varieties that become isogenous to a power of E overa Galois extension. This functor may therefore shed light upon abelianvarieties that are not isogenous to a power of E over the ground field, andtherefore missed by the previous functor, but become isogenous to a powerof E after making a suitable extension.Let k′/k be a finite Galois extension and let G := Gal(k′/k). As everyproper commutative group scheme over a field is projective, the categoryof commutative proper k-group schemes is equivalent to the category ofcommutative proper k′-group schemes equipped with descent data for k′/k.For this reason we may identify commutative proper k-group schemes withcommutative proper k′-group schemes with an action of G, such that allmaps, including the structure maps, are G-equivariant.Let R = Rk′ := EndEk′ be the endomorphism ring of the base changeEk′ of E to k′. In particular, R is noetherian. This inherits an action of G.We denote this action by r 7→ σr for σ ∈ G and r ∈ R. We may then formthe twisted group ring R〈G〉 as the free R-moduleR〈G〉 =⊕σ∈GR · σwith the commutation relation σr = σrσ. In this way, a module over R〈G〉is an R-module with a semilinear G-action.We give a construction of the following in Section 2.Abelian varieties isogenous to a power of an elliptic curve 207Theorem 1.1. There exists an exact functorHomR〈G〉( · , Ek′) :{fin. pres. leftR〈G〉-modules}opp→{commutative properk-group schemes},such that(1) For a finitely presented R〈G〉-module M and any k-scheme C, wehaveHomk(C,HomR〈G〉(M,Ek′)) = HomR〈G〉(M,Homk′(Ck′ , Ek′)).(2) The functor agrees with HomR( · , Ek′) under the forgetful functorsmapping R〈G〉-modules to R-modules and base-changing k-schemesto k′-schemes.In particular, this shows that if M is an R〈G〉-module that is torsion-free as an R-module, then HomR〈G〉(M,Ek′) is an abelian variety A over k,such that Ak′ is isogenous to a power of E. However, we may say somethingabout the k-isogeny class of A as well. Recall that the category of abelianvarieties over k up to isogeny has the same objects as the category of abelianvarieties over k, but all isogenies are inverted.Let F = R⊗ZQ. Since F is semisimple, the category of finitely presentedmodules over the twisted group algebra F 〈G〉 is semisimple (as in the casewhen the action of G on F is trivial, see Lemma 2.3 below). Let S1, . . . , S`denote the simple objects.Theorem 1.2. There is a functorHomF 〈G〉( · , Ek′) :{fin. pres.F 〈G〉-modules}opp→{abelian varieties over kup to isogeny},compatible with the functor HomR〈G〉( · , Ek′). For any torsion-free finitelypresented R〈G〉-module M , HomR〈G〉(M,Ek′) is isogenous over k to aproduct of powers of the abelian varieties HomF 〈G〉(Si, Ek′).As an application of this Galois-equivariant functor, we give a simple andnew proof of the following old result (see [3, Prop. 25], [2, Prop. 2.2], [6,Prop. 5.3], [5, Prop. 2.3b] for other proofs of the same or similar results),which is useful in reducing questions about arbitrary CM elliptic curves tothose with complex multiplication by a maximal order; this will be usedin [7] for this very reason.Corollary 1.3. Let k be a number field and let E/k be an elliptic curve.Suppose that Ek̄ has complex multiplication by an order O in an imaginaryquadratic field F . Then there exists an elliptic curve E′/k and an isogenyϕ : E′ → Edefined over k, such that E′k̄has complex multiplication by the full ring ofintegers of F .208 Isabel VogtAcknowledgments. I would like to thank Bjorn Poonen for suggestingthat the initial functor could be useful in addressing the problem in Corol-lary 1.3 and for helpful discussions. I also thank Pavel Etingof for answeringsome questions, and Pete Clark for explaining the history of the result inCorollary 1.3 and providing references.2. Categorical ConstructionsFor any R〈G〉-module M , let HomR〈G〉(M,Ek′) be the functor from thecategory of k-schemes to the category of sets sending a k-scheme C toHomR〈G〉(M,Homk′(Ck′ , Ek′)).LetResk′/k : {comm. proper k′-group schemes}→{comm. proper k-group sch}denote Weil restriction of scalars.Proposition 2.1. The functor HomR〈G〉(M,Ek′) is representable by acommutative proper k-group scheme. When M = R〈G〉, the representingscheme is Resk′/k Ek′.Proof. We first show that HomR〈G〉(R〈G〉, Ek′) = Resk′/k Ek′ . Indeed, bythe universal property of the restriction of scalars we haveHomk(C,Resk′/k Ek′)= Homk′(Ck′ , Ek′) = HomR〈G〉(R〈G〉,Homk′(Ck′ , Ek′))= Homk(C,HomR〈G〉(R〈G〉, Ek′))as desired. Note that the k-scheme Resk′/k Ek′ has endomorphisms by R〈G〉.Now let(2.1) R〈G〉m → R〈G〉n →M → 0be a finite presentation of M as R〈G〉-modules. The map R〈G〉m → R〈G〉nis given by multiplication on the right by a m×n matrix X. Multiplicationby X on the left also defines a map(Resk′/k Ek′)n → (Resk′/k Ek′)m.As commutative proper group schemes over k form an abelian category, theabove morphism has a kernel in this category,(2.2) 0→ A→ (Resk′/k Ek′)n → (Resk′/k Ek′)m.We claim that A represents the functor HomR〈G〉(M,Ek′) defined above.Indeed, we may apply the left-exact functor Homk(C,−) to (2.2) to obtain(2.3) 0→ Homk(C,A)→ Homk′(Ck′ , Ek′)n → Homk′(Ck′ , Ek′)m.Abelian varieties isogenous to a power of an elliptic curve 209Similarly apply the left-exact functor HomR〈G〉( · ,Homk′(Ck′ , Ek′)) to (2.1)to obtain(2.4) 0→ HomR〈G〉(M,Homk′(Ck′ , Ek′))→ Homk′(Ck′ , Ek′)n→ Homk′(Ck′ , Ek′)m.As the right maps in both (2.3) and (2.4) are induced by multiplication byX, the kernels are functorially isomorphic, as desired. Recall that we let F := R ⊗ Q. We now show that we have a compati-ble functor taking values in the category of abelian varieties over k up toisogeny, as in Theorem 1.2. For objects B1 and B2 in the isogeny category,denote by Homisog(B1, B2) the morphisms in the isogeny category.For a finitely presented F 〈G〉-module N , define the functorHomF 〈G〉(N,Ek′)from the isogeny category of abelian varieties over k to sets byB 7→ HomF 〈G〉(N,Homisog(Bk′ , Ek′)).Proposition 2.2. The functor HomF 〈G〉(N,Ek′) defined above is repre-sented by an abelian variety in the isogeny category over k.Proof. As above, let B be an object of the isogeny category of abelianvarieties over k. If N = M ⊗R〈G〉 F 〈G〉, we haveHomF 〈G〉(N,Homisog(Bk′ , Ek′))= HomF 〈G〉(M ⊗ F 〈G〉,Homk′(Bk′ , Ek′)⊗Q),= HomR〈G〉(M,R〈G〉Homk′(Bk′ , Ek′)⊗Q),= HomR〈G〉(M,Homk′(Bk′ , Ek′))⊗Q.And so this functor agrees with the original HomR〈G〉( · , Ek′) after com-posing with the localization map to the isogeny category.Therefore the functor HomF 〈G〉(F 〈G〉, Ek′) is represented by Resk′/k(Ek′)in the isogeny category. Furthermore, as the isogeny category is an abeliancategory, the same proof as in Lemma 2.1 shows that for all finitely pre-sented F 〈G〉-modules N , the functor HomF 〈G〉(N,Ek′) is represented byan object in the isogeny category. Lemma 2.3. Let F be a semisimple Q-algebra. Then F 〈G〉 is semisimple.Proof. Let V →W be a surjection of finite-dimensional left F 〈G〉-modules.As F is semisimple, we may choose a splitting ϕ : W → V as F -modules.The map π : W → V given byπ(w) =∑g∈Ggϕ(g−1w),defines a splitting as F 〈G〉-modules. 210 Isabel VogtSince the endomorphism algebra of an elliptic curve is always semisimple,Lemma 2.3 combined with the above construction proves Theorem 1.2.We now show compatibility with base change and restriction of scalars.Lemma 2.4. Let M be a finitely presented left R-module and set A :=HomR(M,Ek′). Then we have thatResk′/k A 'HomR〈G〉(R〈G〉 ⊗RM,Ek′).Proof. By Yoneda’s Lemma, it suffices to show that Resk′/k A andHomR〈G〉(R〈G〉 ⊗RM,Ek′have the same functors of points. Let C be a k-scheme. By the universalproperty of restriction of scalars, we haveResk′/k A(C) = A(Ck′) = HomR(M,Hom(Ck′ , Ek′)).By the adjunction between restriction and induction [1, Prop. 2.8.3(i)], wehaveHomR(M,Hom(Ck′ , Ek′)) ' HomR〈G〉(R〈G〉 ⊗M,Hom(Ck′ , Ek′)),which completes the proof. For any ring R and any left R-algebra S, let S denote the (R,S)-bimoduleS under multiplication by R on the left and S on the right.Lemma 2.5. Let M be a finitely presented left R〈G〉-module and let A :=HomR〈G〉(M,Ek′). Then the base change Ak′ is isomorphic toHomR(RM,Ek′),where RM denotes the underlying R-module of M .Proof. By Yoneda’s Lemma, it suffices to show that Ak′ and HomR(RM,Ek′) have the same functor of points. Let D be a k′-scheme. Let kD denotethe k-scheme whose structure morphism is the composition D → Spec k′ →Spec k. By the universal property of fiber products,Ak′(D) = A(kD) = HomR〈G〉(M,Ek′((kD)k′)).We are thus reduced to showing thatHomR〈G〉(M,Ek′((kD)k′))= HomR(RM,Ek′(D)).Furthermore, by the adjunction between restriction and coinduction [1,Prop. 2.8.3(ii)],HomR(RM,Ek′(D)) = HomR〈G〉(M,HomR(R〈G〉, Ek′(D))).It suffices then to show that, as left R〈G〉-modules,Ek′((kD)k′)' HomR(R〈G〉, Ek′(D)).Abelian varieties isogenous to a power of an elliptic curve 211As k′ ⊗k k′ '∏σ∈G k′, we have that(kD)k′ ' qσ∈GD,where τ ∈ G maps the σth copy of D to the τσth copy. Elements ofEk′ (qσ∈GD) can be represented as tuples f = (fσ)σ where fσ ∈ Ek′(D)and(τf)σ = τfτ−1σ.Similarly elements of HomR(R〈G〉, Ek′(D)) are tuples g = (gσ)σ whereg ∈ Ek′(D) and (τg)σ = gστ . The correspondencegσ = σfσ−1 ,shows that these R〈G〉-modules are isomorphic. Note that any finitely presented R-module with a semilinear G-action isalso a finitely presented R〈G〉-module, as we now explain. As R is noether-ian, it suffices to show that any R〈G〉-moduleM , which is finitely generatedas an R-module, admits a G-equivariant surjection from R〈G〉n for some n.If m1, . . . ,mr are R-module generators of M , then the map R〈G〉r → Msending ei 7→ mi has the desired properties.We have the following nice properties of the functor HomR〈G〉( · , Ek′).Proposition 2.6. Let E/k be an elliptic curve and k′/k a finite Galoisextension with Galois group G. Let R := Endk′ E and let M be a finitelypresented R〈G〉-module and let A := HomR〈G〉(M,Ek′).(1) HomR〈G〉( · , Ek′) is exact.(2) A is a commutative proper group scheme over k of dimension rkR(M).(3) If M is torsion-free as an R-module, then A is an abelian varietyover k such that Ak′ is isogenous to a power of Ek′.(4) HomR〈G〉(R,Ek′) = E.Proof. Parts (1)–(3) follow from the corresponding properties after base-extension to k′ [4, ThM. 4.4]. For part (4), we haveHomk(C,HomR〈G〉(R,Ek′)) = HomR〈G〉(R,Homk′(Ck′ , Ek′))= Homk′(Ck′ , Ek′)G= Homk(C,E). 3. Examples and ApplicationsAs an example of Theorem 1.2, we have the following in the special casek′/k is a separable quadratic extension.Proposition 3.1. Let k′/k be a separable quadratic extension and set G :=Gal(k′/k) with nontrivial element σ. Let E′ denote the corresponding k′/kquadratic twist of E. If M is a torsion-free R〈G〉-module, then the abelianvariety HomR〈G〉(M,Ek′) is isogenous to Er × (E′)r′for some r, r′ ≥ 0.212 Isabel VogtProof. As above, we denote the G-action on F by σ(x) = σx. By Theo-rem 1.2, it suffices to decompose F 〈G〉 into simple modules. Let σF denoteF endowed with the G-action σ(x) = −σx for x ∈ F . Then we have anisomorphismF ⊕ σF ϕ−→ F 〈G〉,defined by ϕ(a, b) = a(σ + 1) + b(σ − 1). By comparing functors of points,we have HomF 〈G〉(F,Ek′) ' E and HomF 〈G〉(σF,Ek′) ' E′ in the isogenycategory. Example 3.2. Consider the special case that k is a number field, andE has complex multiplication by an order O in the imaginary quadraticfield F = FracO, which is defined over the quadratic extension k′ = kF/kwith Galois group G = {1, σ}. Let M be a finitely presented O〈G〉-modulethat is torsion-free as an O-module. Then as multiplication by a totallyimaginary element of O defines an isomorphism σF ' F of F 〈G〉-modules,we have that HomO〈G〉(M,Ek′) is an abelian variety defined over k, whichis isogenous over k to a power of E.Continuing in the setup of the previous example, we conclude by provingCorollary 1.3.Proof of Corollary 1.3. This follows from Example 3.2 as the full ring ofintegers OF of F is a finitely presented O〈G〉-module, which is torsion-free of rank 1 over O. The full endomorphism ring of E is defined over(the at most quadratic extension) k′, so we have that O = EndEk′ . Thisinherits a (possibly trivial) action of G. We therefore have the followingexact sequence of O〈G〉-modules:0→ O → OF → OF /O → 0,where the action of G is induced from the action on F . Applying the functorHomO〈G〉( · , Ek′) we obtain an exact sequence0→HomO〈G〉(OF /O, Ek′)→ E′ → E → 0,where E′ is again an elliptic curve over k and the right map is an isogenyto E. By functoriality, E′k′ has an action of OF , as desired. References[1] D. J. Benson, Representations and cohomology. I. Basic representation theory of finitegroups and associative algebras, Cambridge Studies in Advanced Mathematics, vol. 30, Cam-bridge University Press, 1998.[2] A. Bourdon & P. Pollack, “Torsion subgroups of CM elliptic curves over odd degreenumber fields”, Int. Math. Res. Not. 2017 (2017), no. 16, p. 4923-4961.[3] P. L. Clark, B. Cook & J. Stankewicz, “Torsion points on elliptic curves with complexmultiplication”, Int. J. Number Theory 9 (2013), no. 2, p. 447-479.Abelian varieties isogenous to a power of an elliptic curve 213[4] B. W. Jordan, A. G. Keeton, B. Poonen, E. M. Rains, N. Shepherd-Barron & J. T.Tate, “Abelian varieties isogenous to a power of an elliptic curve”, Compos. Math. 154(2018), no. 5, p. 934-959.[5] S. Kwon, “Degree of isogenies of elliptic curves with complex multiplication”, J. KoreanMath. Soc. 36 (1999), no. 5, p. 945-958.[6] K. Rubin, “Elliptic curves with complex multiplication and the conjecture of Birch andSwinnerton–Dyer”, in Arithmetic theory of elliptic curves (Cetraro, 1997), Lecture Notes inMathematics, vol. 1716, Springer, 1997, p. 167-234.[7] I. Vogt, “A local-global principle for isogenies of composite degree”, https://arxiv.org/abs/1801.05355, 2018.Isabel VogtDepartment of MathematicsMassachusetts Institute of Technology77 Massachusetts AvenueCambridge, MA 02139, USAE-mail: ivogt@mit.eduURL: http://web.mit.edu/~ivogt/

Abelian varieties isogenous to a power of an elliptic curve over a Galois extension

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Abelian varieties isogenous to a power of an elliptic curve over a Galois extension

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