We give a characterization of ellptic curves which are isogenous over two different quadratic extensions of a fixed number field

Ivica Gusić

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GLASNIK MATEMATIČKIVol. 35(55)(2000), 335 – 337ON ISOGENIES OF ELLIPTIC CURVESIvica GusićUniversity of Zagreb, CroatiaAbstract. We give a characterization of elliptic curves which areisogenous over two different quadratic extensions of a fixed number field.1. Main ResultLet k be a number field and let A,B be elliptic curves over k. An isogenybetween A and B is a nonzero morphism f : A→ B which is a homomorphismof groups. We say that A,B are isogenous if there is an isogeny f : A → B.If f is defined over an extension field K/k then we say that the elliptic curvesare isogenous over K (or K-isogenous). The set of isogenies from A to itself,including the zero map, form a ring End(A) which is called the endomorphismring of A. An elliptic curve A is said to have complex multiplication (CMfor short) if its endomorphism ring End(A) is strictly larger then Z. If anelliptic curve A defined over a number field has CM, then the algebra K0 =End(A)⊗ZQ is a quadratic imaginary field over rationals. For basic definitionsand results on elliptic curves see [1] and [3].Let A be an elliptic curve over k defined by a Weierstrass equation y2 =f(x) and let d ∈ k be a nonsquare. By Ad we denote the elliptic curve withequation dy2 = f(x). It is easy to see that A and Ad are isomorphic overk(√d).Let k be a number field and let A,B be elliptic curves over k withoutcomplex multiplication. Then there exists quadratic extension K/k such thatevery isogeny f : A → B is defined over K (see [2, Remark 5.8]). Therefore,if A,B are isogenous over two different quadratic extensions of k, than A,Bare isogenous over k. The following example shows that this is not true forelliptic curves with complex multiplication.2000 Mathematics Subject Classification. 14H52.Key words and phrases. elliptic curves with complex multiplication, isogeny.335336 IVICA GUSIĆExample 1.1. Let A be the elliptic curve with Weierstreiss equationy2 = x3 + 1 and let A3 be the elliptic curve with equation 3y2 = x3 + 1.These elliptic curves have complex multiplication with the field of complexmultiplication K0 = Q(√−3).Put α(x, y) = (x, y√3). Then α : A → A3 is an isomorphism definedover Q(√3). Similarly β : A → A, β(x, y) = (ρx, y), where ρ is a nontrivialthird root of unity, is an isogeny defined over Q(√−3). By [3, III, sect.9]the ring End(A) is an order in a quadratic imaginary field. Since Z[β] is amaximal order in Q(√−3), we see that End(A) is generated as Z-module by1 and β, so any endomorphism of A is defined over Q(√−3). Let σ be thenontrivial automorphism of Q(√3,√−3) over Q(√−1). Then (α◦(2β+1))σ =−α ◦ (−2β − 1) = α ◦ (2β + 1), so A and A3 are isogenous both over Q(√3)and over Q(√−1).Let φ : A→ A3 be an isogeny. Then φ = α ◦ (α−1 ◦φ). Since α is definedover Q(√3), and α−1 ◦ φ is defined over Q(√−3), we see that φ is definedover Q(√−3,√3). Let σ be the nontrivial automorphism of Q(√3,√−3) overQ(√−3). Then φσ = (α◦(α−1◦φ))σ = −α◦(α−1◦φ) = −φ. Therefore, A andA3 are not isogenous over Q (in fact they are not isogenous over Q(√−3)).We will show that the example is typical.Theorem 1.2. Let k be a number field, let K = k(√d), M = k(√D) bedifferent quadratic extension fields of k, for d,D ∈ k. Then:(a) If A,B are elliptic curves over k such that A,B are isogenous bothover K and over M and such that they are not isogenous over k, then:(i) B and Ad are isogenous over k.(ii) A and B have complex multiplication.(iii) k(√dD) = kK0, where K0 is the corresponding field of complexmultiplication.(b) If k(√dD) = kK0, where K0 is a quadratic imaginary field over Q, andif A is an elliptic curve over k with the field of complex multiplicationK0, then A,Ad are isogenous both over K and over M , but they arenot isogenous over k.Proof. a) (i) Assume A is defined by a Weierstrass equation y2 = f(x).Let φ : B → A be an isogeny defined over K and not defined over k, and letσ be the nontrivial automorphism of K over k. Then φ − φσ : B → A is anisogeny. Therefore α ◦ (φ − φσ) : B → Ad, where α : A → Ad is defined byα(x, y) = (x, y√d), is an isogeny defined over k.(ii) Follows directly from [2, Remark 5.8](iii) Let φd : A → B be a K-isogeny and let φD : B → A be an M -isogeny.Then ψ = φD ◦ φd : A → A, is defined over KM . By [1, 5.1.3] ψ is definedover kK0. Denote by σ the nontrivial automorphism of KM/k which is trivialon M and by τ the nontrivial automorphism of KM/k which is trivial on K.ON ISOGENIES OF ELLIPTIC CURVES 337By the proof of a) (i), we see that φd,φD can be chosen such that φdσ = −φdand φDτ = −φD. Therefore ψσ = −ψ, so ψ is not defined over M . Since ψ isdefined over kK0, we see that kK0 is quadratic over k, and since ψ is definedover KM , we see that kK0 ⊆ KM (in contrary ψ would be defined over k).Also, ψτ = −ψ, so ψ is not defined over K nor over M . Since ψ is definedover kK0 we see that kK0 is different both from K and from M . ThereforekK0 = k(√dD).b) Let α be as above, let σ be the nontrivial automorphism of KM/kwhich is trivial on M and let τ be the nontrivial automorphism of KM/kwhich is trivial on K. Choose any Φ ∈ EndA \Z. By [1, 5.1.3 and 5.1.1] Φ isdefined over kK0, but Φ is not defined over k. Since σ is nontrivial on kK0,Ψ = α ◦ (Φ − Φσ) : A → Ad is an isogeny defined over KM . In fact, Ψ isdefined over M , because Ψσ = (−α) ◦ (Φσ −Φ) = Ψ. Therefore A and Ad areisogenous both over K and over M .Let us prove that A and Ad are not isogenous over k. Moreover, wewill prove that A and Ad are not isogenous over kK0. Let θ : A → Adbe any isogeny. Then θ = α ◦ (α−1 ◦ θ). By [1, 5.1.3] α−1 ◦ θ is definedover kK0, so since kK0 ⊆ KM we see that θ is defined over KM . Sinceθστ = (−α) ◦ (α−1 ◦ θ) = −θ, θ is not defined over kK0.Corollary 1.3. Let A,B be elliptic curves defined over a number fieldk. If A and B are isogenous over three different quadratic extensions over k,then they are isogenous over k.Remark 1.4. In the first version of the paper, the proof of the theoremhas been based on the Faltings theorem on isogenies of abelian varieties. Thepresent proof is more elementary and is suggested by the referee.References[1] G.Shimura, Introduction to the Arithmetic Theory of Automorphic Functions,Iwanami Shoten Publishers and Princeton University Press, 1971.[2] A.Silverberg, Fields of definition for homomorphism of abelian varieties, Journal ofPure and Applied Algebra, 77, No3 (1992)253–264.[3] J.Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 106,1986.University of ZagrebFaculty of Chemical Engineering and Technologypp. 177, 10 001 ZagrebCroatiaE-mail : igusic@pierre.fkit.hrReceived : 24.09.1998.Revised : 22.11.1999.Revised : 26.07.2000.

English

On isogenies of elliptic curves

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GLASNIK MATEMATICKIVol. 35(55)(2000), 335 { 337ON ISOGENIES OF ELLIPTIC CURVESIvica GusicUniversity of Zagreb, CroatiaAbstract. We give a characterization of elliptic curves which areisogenous over two dierent quadratic extensions of a xed number eld.1. Main ResultLet k be a number eld and let A;B be elliptic curves over k. An isogenybetween A and B is a nonzero morphism f : A! B which is a homomorphismof groups. We say that A;B are isogenous if there is an isogeny f : A ! B.If f is dened over an extension eld K=k then we say that the elliptic curvesare isogenous over K (or K-isogenous). The set of isogenies from A to itself,including the zero map, form a ring End(A) which is called the endomorphismring of A. An elliptic curve A is said to have complex multiplication (CMfor short) if its endomorphism ring End(A) is strictly larger then Z. If anelliptic curve A dened over a number eld has CM, then the algebra K0 =End(A)ZQ is a quadratic imaginary eld over rationals. For basic denitionsand results on elliptic curves see [1] and [3].Let A be an elliptic curve over k dened by a Weierstrass equation y2 =f(x) and let d 2 k be a nonsquare. By Ad we denote the elliptic curve withequation dy2 = f(x). It is easy to see that A and Ad are isomorphic overk(pd).Let k be a number eld and let A;B be elliptic curves over k withoutcomplex multiplication. Then there exists quadratic extension K=k such thatevery isogeny f : A! B is dened over K (see [2, Remark 5.8]). Therefore,if A;B are isogenous over two dierent quadratic extensions of k, than A;Bare isogenous over k. The following example shows that this is not true forelliptic curves with complex multiplication.2000 Mathematics Subject Classication. 14H52.Key words and phrases. elliptic curves with complex multiplication, isogeny.335336 IVICA GUSICExample 1.1. Let A be the elliptic curve with Weierstreiss equationy2 = x3 + 1 and let A3 be the elliptic curve with equation 3y2 = x3 + 1.These elliptic curves have complex multiplication with the eld of complexmultiplication K0 = Q(p 3).Put (x; y) = (x; yp3). Then  : A ! A3 is an isomorphism denedover Q(p3). Similarly  : A ! A, (x; y) = (x; y), where  is a nontrivialthird root of unity, is an isogeny dened over Q(p 3). By [3, III, sect.9]the ring End(A) is an order in a quadratic imaginary eld. Since Z[] is amaximal order in Q(p 3), we see that End(A) is generated as Z-module by1 and , so any endomorphism of A is dened over Q(p 3). Let  be thenontrivial automorphism of Q(p3;p 3) over Q(p 1). Then ((2+1)) =   ( 2   1) =   (2 + 1), so A and A3 are isogenous both over Q(p3)and over Q(p 1).Let  : A! A3 be an isogeny. Then  =   ( 1 ). Since  is denedover Q(p3), and  1   is dened over Q(p 3), we see that  is denedover Q(p 3;p3). Let  be the nontrivial automorphism of Q(p3;p 3) overQ(p 3). Then  = (( 1)) =  ( 1) =  . Therefore, A andA3 are not isogenous over Q (in fact they are not isogenous over Q(p 3)).We will show that the example is typical.Theorem 1.2. Let k be a number eld, let K = k(pd), M = k(pD) bedierent quadratic extension elds of k, for d;D 2 k. Then:(a) If A;B are elliptic curves over k such that A;B are isogenous bothover K and over M and such that they are not isogenous over k, then:(i) B and Ad are isogenous over k.(ii) A and B have complex multiplication.(iii) k(pdD) = kK0, where K0 is the corresponding eld of complexmultiplication.(b) If k(pdD) = kK0, where K0 is a quadratic imaginary eld over Q, andif A is an elliptic curve over k with the eld of complex multiplicationK0, then A;Ad are isogenous both over K and over M , but they arenot isogenous over k.Proof. a) (i) Assume A is dened by a Weierstrass equation y2 = f(x).Let  : B ! A be an isogeny dened over K and not dened over k, and let be the nontrivial automorphism of K over k. Then     : B ! A is anisogeny. Therefore   (   ) : B ! Ad, where  : A ! Ad is dened by(x; y) = (x; ypd), is an isogeny dened over k.(ii) Follows directly from [2, Remark 5.8](iii) Let d : A! B be a K-isogeny and let D : B ! A be an M -isogeny.Then  = D  d : A ! A, is dened over KM . By [1, 5.1.3]  is denedover kK0. Denote by  the nontrivial automorphism of KM=k which is trivialon M and by  the nontrivial automorphism of KM=k which is trivial on K.ON ISOGENIES OF ELLIPTIC CURVES 337By the proof of a) (i), we see that d,D can be chosen such that d =  dand D =  D. Therefore   =   , so  is not dened over M . Since  isdened over kK0, we see that kK0 is quadratic over k, and since  is denedover KM , we see that kK0  KM (in contrary  would be dened over k).Also,   =   , so  is not dened over K nor over M . Since  is denedover kK0 we see that kK0 is dierent both from K and from M . ThereforekK0 = k(pdD).b) Let  be as above, let  be the nontrivial automorphism of KM=kwhich is trivial on M and let  be the nontrivial automorphism of KM=kwhich is trivial on K. Choose any  2 EndA nZ. By [1, 5.1.3 and 5.1.1]  isdened over kK0, but  is not dened over k. Since  is nontrivial on kK0,	 =   (   ) : A ! Ad is an isogeny dened over KM . In fact, 	 isdened over M , because 	 = ( )  (  ) = 	. Therefore A and Ad areisogenous both over K and over M .Let us prove that A and Ad are not isogenous over k. Moreover, wewill prove that A and Ad are not isogenous over kK0. Let  : A ! Adbe any isogeny. Then  =   ( 1  ). By [1, 5.1.3]  1   is denedover kK0, so since kK0  KM we see that  is dened over KM . Since = ( )  ( 1  ) =  ,  is not dened over kK0.Corollary 1.3. Let A,B be elliptic curves dened over a number eldk. If A and B are isogenous over three dierent quadratic extensions over k,then they are isogenous over k.Remark 1.4. In the rst version of the paper, the proof of the theoremhas been based on the Faltings theorem on isogenies of abelian varieties. Thepresent proof is more elementary and is suggested by the referee.References[1] G.Shimura, Introduction to the Arithmetic Theory of Automorphic Functions,Iwanami Shoten Publishers and Princeton University Press, 1971.[2] A.Silverberg, Fields of denition for homomorphism of abelian varieties, Journal ofPure and Applied Algebra, 77, No3 (1992)253{264.[3] J.Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 106,1986.University of ZagrebFaculty of Chemical Engineering and Technologypp. 177, 10 001 ZagrebCroatiaE-mail : igusic@pierre.fkit.hrReceived : 24.09.1998.Revised : 22.11.1999.Revised : 26.07.2000.

On isogenies of elliptic curves

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