A local-global principle for isogenies of prime degree over number fields
aa r X i v : . [ m a t h . N T ] J a n A local-global principle for isogenies of prime degreeover number fields
Samuele AnniNovember 4, 2018
Abstract
We give a description of the set of exceptional pairs for a numberfield K , that is the set of pairs ( ℓ, j ( E )), where ℓ is a prime and j ( E ) isthe j -invariant of an elliptic curve E over K which admits an ℓ -isogenylocally almost everywhere but not globally. We obtain an upper boundfor ℓ in such pairs in terms of the degree and the discriminant of K .Moreover, we prove finiteness results about the number of exceptionalpairs. Let E be an elliptic curve over a number field K , and let ℓ be a primenumber. If we know residual information on E , i.e. information over thereduction of E for a set of primes with density one, can we deduce globalinformation about E ?One of the first to ask this kind of question was Serge Lang: let E be anelliptic curve over a number field K , and let ℓ be a prime numbe; if E hasnon-trivial ℓ -torsion locally at a set of primes with density one , then does E have non-trivial ℓ -torsion over K ? In 1981, Katz studied this local-globalprinciple, see [Kat81]. He was able to prove that if E has non-trivial ℓ -torsionlocally at a set of primes with density one, then there exists a K -isogenouselliptic curve which has non-trivial ℓ -torsion over K . He proved this byreducing the problem to a purely group-theoretic statement. Moreover, hewas able to extend the result even further to 2-dimensional abelian varietiesand to give a family of counterexamples in dimension 3.Here we consider the following variation on this question: let E be anelliptic curve over a number field K , and let ℓ be a prime number; if E admits an ℓ -isogeny locally at a set of primes with density one then does E admit an ℓ -isogeny over K ?Recently, Sutherland has studied this problem, see [Sut12]. Let us recallthat, except in the case when the j -invariant is 0 or 1728, whether an ellipticcurve admits an ℓ -isogeny over K or not depends only on its j -invariant. As1n [Sut12], we will only consider elliptic curves with j -invariant differentfrom 0 and 1728. Definition 1.1.
Let K be a number field and E an elliptic curve over K .A pair ( ℓ, j ( E )) is said to be exceptional for K if E/K admits an ℓ -isogenylocally almost everywhere, i.e. for a set of primes of K of density one, butnot over K . For primes of good reduction and not dividing ℓ , the definition of localisogeny is the natural one , and it is recalled in Definition 2.1 below.Let us remark that if ( ℓ, j ( E )) is an exceptional pair for the number field K , then any E D , quadratic twist of E , gives rise to the same exceptionalpair. Indeed, the Galois representation associated to the ℓ -torsion of E andthe one of E D are twists of each other: ρ E D ,ℓ ≃ χ D ⊗ ρ E,ℓ , where χ D isa quadratic character. Hence the projective images of such representationsare isomorphic.A curve occurring in an exceptional pair will admit an ℓ -isogeny globallyover a small extension of the base field: more precisely, we can state thefollowing Proposition, which is a sharpened version of a result of Sutherland(for a proof see Section 3 herein): Proposition 1.2.
Let E be an elliptic curve defined over a number field K , let ℓ be an odd prime number and assume that q(cid:0) − ℓ (cid:1) ℓ does not belongto K . Suppose that E/K admits an ℓ -isogeny locally at a set of primeswith density one. Then E admits an ℓ -isogeny over K ( √− ℓ ) . Moreover, if ℓ = 2 , or ℓ ≡ then E admits an ℓ -isogeny over K . There are examples, for ℓ ≡ ℓ ≥
7, in which it is necessaryto extend the base field to have a global isogeny. In particular, Sutherlandproved that over Q the following optimal result holds: Theorem 1.3 ([Sut12, Theorem 2]) . The only exceptional pair for Q is (7 , / . This theorem is proved by applying [Par05, Theorem 1.1], which assertsthat for all primes ℓ ≡ ℓ >
7, the only rational non-cuspidalpoints on X split ( ℓ )( Q ) correspond to elliptic curves with complex multipli-cation (for a definition of this modular curve see Section 5 herein). Hence,over Q there exists only one counterexample to the local-global principle for7-isogenies and there is none for ℓ -isogenies for ℓ > ℓ -isogeniesfor ℓ > ain Theorem. Let K be a number field of degree d over Q and discrim-inant ∆ , and let ℓ K := max {| ∆ | , d +1 } . The following holds: (1) if ( ℓ, j ( E )) is an exceptional pair for the number field K then ℓ ≤ ℓ K ; (2) there are only finitely many exceptional pairs for K with < ℓ ≤ ℓ K ; (3) the number of exceptional pairs for K with ℓ = 7 is finite or infinite,depending whether the rank over K of the Elkies-Sutherland’s ellipticcurve: y = x − x + 33614 is zero or non-zero, respectively; (4) there exist no exceptional pairs for K with ℓ = 2 or with ℓ = 3 ; (5) there exist exceptional pairs for K with ℓ = 5 if and only if √ belongsto K . Moreover, if √ belongs to K then there are infinitely manyexceptional pairs for K with ℓ = 5 . Actually, we prove more precise results that will be discussed in thefollowing sections. Before entering the details of the proofs, let us give arough idea of our strategy for the point (1) above.The pair ( ℓ, j ( E )) is exceptional for a number field K if and only ifthe action of G ⊆ GL ( F ℓ ), the image of the Galois representation as-sociated to the ℓ -torsion of E , on P ( E [ ℓ ]) ≃ P ( F ℓ ) has no fixed point,whereas every g ∈ G leaves a line stable, that is, all g ∈ G have a re-ducible characteristic polynomial. Using Dickson’s classification of sub-groups of PGL ( F ℓ ), one sees that, up to conjugation, G has to be eitherthe inverse image of an exceptional group (but this case is known to happenonly for small ℓ ) or contained in the normalizer of a split Cartan subgroup,that is G = (cid:26)(cid:18) a b (cid:19) , (cid:18) ab (cid:19) | a, b ∈ Γ (cid:27) for Γ a subgroup of F ∗ ℓ .In the last case, using notations as in [Par05, Section 2] or in Section 5herein, exceptional pairs therefore induce points in X split ( ℓ )( K ) not liftingto X sp . Car ( ℓ )( K ) (forgetting cases corresponding to exceptional groups). Theexistence of such points is in general a wide open question; however, inour case, the reducibility of the characteristic polynomials implies that Γis actually a subgroup of squares: Γ ⊆ ( F ∗ ℓ ) . By the property of the Weilpairing, one deduces that K ( q(cid:0) − ℓ (cid:1) ℓ ) ⊆ L , the field of definition of the ℓ -isogeny. From this, in the case q(cid:0) − ℓ (cid:1) ℓ does not belong to K , we concludethat the well-known shape of inertia at ℓ inside G gives a contradictionfor ℓ > K : Q ]+1.This article is organized as follows. For the convenience of the reader, werecall in Section 2 the results obtained by Sutherland in [Sut12, Section 2].In Section 3, we study exceptional pairs over arbitrary number fields. Firstwe deduce the effective version of Sutherland’s result, then we describe how3o tackle the case not treated by Sutherland and finally we prove statement(4) of the Main Theorem (see Proposition 3.9). In Section 4, we prove, aswe already commented, the bound given in (1) of the Main Theorem (Corol-lary 4.5). In Section 5 we discuss finiteness results for the set of exceptionalpairs and we prove statements (2) (Theorem 5.3), (5) (Corollary 5.6) and(3) (Proposition 5.8) of the Main Theorem. Finally, in Section 6, we giveconditions under which an exceptional pair does not have complex multipli-cation.After this article was written, Banwait and Cremona have presented a de-tailed description of the local-global principle about ℓ -isogenies for quadraticnumber fields, see [BC13]. Let us recall the definition of local ℓ -isogeny for an elliptic curve: Definition 2.1.
Let E be an elliptic curve over a number field K , let ℓ bea prime number. If p is a prime of K where E has good reduction, p notdividing ℓ , we say that E admits an ℓ -isogeny locally at p if the reductionof E modulo p admits an ℓ -isogeny defined over the residue field at p . Let us remark that for a prime p of K where E has good reduction, p not dividing ℓ , the definition given is equivalent to saying that the N´eronmodel of E over the ring of integers of K p admits an ℓ -isogeny. This followsessentially because the ℓ -isogeny in this case is ´etale.Sutherland has proved, under certain conditions, that for an elliptic curvedefined over a number field, the existence of local ℓ -isogenies for a set ofprimes with density one implies the existence of a global ℓ -isogeny: Theorem 2.2 ([Sut12, Theorem 1]) . Let E be an elliptic curve over a num-ber field K with j ( E ) / ∈ { , } , and let ℓ be a prime number. Assume that q(cid:0) − ℓ (cid:1) ℓ / ∈ K , and suppose that E/K admits an ℓ -isogeny locally at a setof primes with density one. Then E admits an ℓ -isogeny over a quadraticextension of K . Moreover, if ℓ ≡ or ℓ < , E admits an ℓ -isogenydefined over K . Let us recall briefly how this theorem is proved. The main tool used isthe theory of Galois representations attached to elliptic curves, see [Ser72],to reduce the problem to a question regarding subgroups of GL ( F ℓ ).There is a natural action of GL ( F ℓ ) on P ( F ℓ ), and the induced actionof PGL ( F ℓ ) is faithful. For an element g of GL ( F ℓ ) or of PGL ( F ℓ ), wewill denote, respectively, by P ( F ℓ ) /g the set of g -orbits of P ( F ℓ ) and by P ( F ℓ ) g the set of elements fixed by g .4 emma 2.3 ([Sut12, Lemma 1]) . Let G be a subgroup of GL ( F ℓ ) whose im-age H in PGL ( F ℓ ) is not contained in SL ( F ℓ ) / {± } . Suppose | P ( F ℓ ) g | > for all g ∈ G but | P ( F ℓ ) G | = 0 .Then ℓ ≡ and the following holds: (1) H is dihedral of order n , where n > is an odd divisor of ( ℓ − / ; (2) G is properly contained in the normalizer of a split Cartan subgroup; (3) P ( F ℓ ) /G , the set of G -orbits of P ( F ℓ ) , contains an orbit of size . This result is an application of the orbit-counting lemma combined withDickson’s classification of subgroups of PGL ( F ℓ ), see [Dic58] or [Lan76],and it is one of the key steps in the proof of Theorem 2.2. Let us noticethat if the hypotheses of Lemma 2.3 are satisfied, then it follows that ℓ = 3because n > Remark . Given an elliptic curve E over a number field K , the compatibil-ity between ρ E,ℓ , the Galois representation associated to the ℓ -torsion group E [ ℓ ], and the Weil pairing on E [ ℓ ] implies that for every σ ∈ Gal( Q /K ) wehave σ ( ζ ℓ ) = ζ det( ρ E,ℓ ( σ )) ℓ , where ζ ℓ is an ℓ th-root of unity. Let us assume that ℓ is an odd prime. Therefore, ζ ℓ is in K if and only if G = ρ E,ℓ (Gal( Q /K )) iscontained in SL ( F ℓ ). Moreover, using the Gauss sum: P ℓ − n =0 ζ n ℓ = q(cid:0) − ℓ (cid:1) ℓ and denoting by H the image of G in PGL ( F ℓ ), it follows that H is con-tained in SL ( F ℓ ) / {± } if and only if q(cid:0) − ℓ (cid:1) ℓ is in K . of Theorem 2.2. For every g ∈ G = ρ E,ℓ (Gal( Q /K )), it follows from Cheb-otarev density theorem that there exists p ⊂ O K , a prime in the ring ofintegers of K , such that g = ρ E,ℓ (Frob p ) and E admits an ℓ -isogeny locallyat p .The Frobenius endomorphism fixes a line in E [ ℓ ], hence | P ( F ℓ ) g | > g ∈ G . If | P ( F ℓ ) G | >
0, then Gal( Q /K ) fixes a linear subspace of E [ ℓ ] which is the kernel of an ℓ -isogeny defined over K .Hence, let us assume | P ( F ℓ ) G | = 0. No subgroup of GL ( F ) satisfies | P ( F ) G | = 0 and | P ( F ) g | > g ∈ G , so ℓ is odd. The hypotheses on K combined with the Weil pairing on the ℓ -torsion implies that some elementof G has a non-square determinant, hence the image of G in PGL ( F ℓ ) doesnot lie in SL ( F ℓ ) / {± } . The hypotheses of Lemma 2.3 are satisfied, thus ℓ ≡ ℓ is different from 3, and P ( F ℓ ) /G contains anorbit of size 2. Let x ∈ P ( F ℓ ) be an element of this orbit, its stabilizeris a subgroup of index 2. By Galois theory, it corresponds to a quadraticextension of K over which E admits an isogeny of degree ℓ (actually, twosuch isogenies). Remark . Since no subgroup of GL ( F ) satisfies | P ( F ) G | = 0 and | P ( F ) g | > g ∈ G , then there is no exceptional pair with ℓ = 2.5 Exceptional pairs and Galois representations
The study of the local-global principle about ℓ -isogenies over an arbitrarynumber field K depends on whether q(cid:0) − ℓ (cid:1) ℓ belongs to K or not. ByRemark 2.5, it follows that there is no exceptional pair with ℓ = 2, hence,unless otherwise stated, we will denote by ℓ an odd prime.First, let us assume that q(cid:0) − ℓ (cid:1) ℓ does not belong to K . Theorem 2.2implies that an exceptional pair for K is no longer exceptional for a quadraticextension of K : in this section we will describe this extension. Proposition 3.1.
Let ( ℓ, j ( E )) be an exceptional pair for the number field K , not containing q(cid:0) − ℓ (cid:1) ℓ , with j ( E ) not in { , } . Let G be the imageof ρ E,ℓ and let H be its image in PGL ( F ℓ ) . Let C ⊂ G be the preimageof the maximal cyclic subgroup of H . Then det( C ) ⊆ ( F ∗ ℓ ) , where ( F ∗ ℓ ) denotes the group of squares in F ∗ ℓ .Proof. Let ( ℓ, j ( E )) be an exceptional pair for the number field K , andassume that q(cid:0) − ℓ (cid:1) ℓ / ∈ K . Therefore, ℓ is odd by Remark 2.5 and Re-mark 2.4 implies that H is not contained in SL ( F ℓ ) / {± } . Hence, applyingLemma 2.3, we have that ℓ ≡ H is a dihedralgroup of order 2 n , where n > ℓ − / G is properly contained in the normalizer of a split Cartan subgroup. Inparticular, we have that ( n, ℓ +1) = 1 and ( n, ℓ ) = 1, because n is odd andit is a divisor of ℓ −
1. Let us underline that, since n is odd the maximalcyclic subgroup inside H is determined uniquely up to conjugation: it is theunique subgroup of index 2.Let A ∈ G ⊂ GL ( F ℓ ) be a preimage of some generator of the maximalcyclic subgroup inside H . Let us prove that A is conjugate in GL ( F ℓ ) to amatrix of the type (cid:18) α β (cid:19) , with α, β ∈ F ∗ ℓ and α/β of order n in F ∗ ℓ .Extending the scalars to F ℓ if necessary, we can put A in its Jordannormal form. Then either A ∼ = (cid:18) α β (cid:19) with α, β ∈ F ℓ , or A ∼ = (cid:18) α α (cid:19) .Since we have A n = λ · Id, for λ ∈ F ∗ ℓ , and ( n, ℓ ) = 1 then the secondcase cannot occur. We claim that α, β ∈ F ∗ ℓ . In fact, let us proceed bycontradiction: if α, β ∈ F ℓ \ F ℓ then β = α , the conjugate of α over F ℓ . Thismeans that P ( F ℓ ) A is empty because A has no eigenvalues in F ℓ and thisis not possible because E admits an ℓ -isogeny locally everywhere and byChebotarev density theorem A = ρ E,ℓ (Frob p ) with p ⊂ O K prime. Hence α, β ∈ F ∗ ℓ . Let us write α = µ i and β = µ j for some generator µ of F ∗ ℓ . Then µ in = α n = β n = µ jn so that µ n ( i − j ) = 1 and n ( j − i ) ≡ ℓ −
1. As n isodd, ( j − i ) has to be even, hence so is ( i + j ). Therefore, det( A ) = αβ = µ i + j is a square in F ∗ ℓ . Moreover, since A is a preimage of some generator of themaximal cyclic subgroup inside H , then α/β must have order n .6 emark . Let ( ℓ, j ( E )) be an exceptional pair for the number field K .Let us suppose that j ( E ) is different from 0 and 1728 and that q(cid:0) − ℓ (cid:1) ℓ isnot in K . Since the projective image of the Galois representation associatedto E is dihedral of order 2 n , with n an odd divisor of ( ℓ − /
2, the order of G , the image of the Galois representation, satisfies: | G | | (( ℓ − · n ) | (cid:18) ( ℓ − · ( ℓ − · (cid:19) = ( ℓ − . Proposition 3.3.
Let ( ℓ, j ( E )) be an exceptional pair for the number field K and assume that q(cid:0) − ℓ (cid:1) ℓ does not belong to K and j ( E ) is different from and . Then E admits an ℓ -isogeny over K ( √− ℓ ) (and actually, twosuch isogenies).Proof. Theorem 2.2 implies that ℓ ≡ ℓ ≥ ℓ, j ( E )) isan exceptional pair. Since √− ℓ / ∈ K , then also ζ ℓ , the ℓ -th root of unity, isnot in K . Since ℓ ≡ Q ( ζ ℓ ) is Q ( √− ℓ ). K ( ζ ℓ ) ❄❄❄⑧⑧⑧⑧⑧ Q ( ζ ℓ ) ⑧⑧⑧⑧⑧ K ( √− ℓ ) ❄❄❄⑧⑧⑧⑧⑧⑧ Q ( √− ℓ ) ⑧⑧⑧⑧⑧ K ❄❄❄❄ Q In particular, Gal( K ( ζ ℓ ) /K ( √− ℓ )) iscontained in ( F ∗ ℓ ) , the subgroup ofsquares inside F ∗ ℓ . If ℓ does not di-vide the discriminant of K then theprevious inclusion is an equality, sincethe ramifications are disjoint. Let, asbefore, ρ E,ℓ : Gal( Q /K ) → GL ( F ℓ )be the Galois representation associ-ated to E and let G be its image.It follows, using notation of Proposition 3.1, that E admits an isogeny(actually two) on the quadratic extension L/K corresponding to the Car-tan subgroup C which is the subgroup of diagonal matrices inside G . ByProposition 3.1, the elements of C have square determinants.On the other hand, from the properties of the Weil pairing on the ℓ -torsion, for all σ ∈ Gal( Q /K ) we have that det( ρ E,ℓ ( σ )) = χ ℓ ( σ ), where χ ℓ is the mod ℓ cyclotomic character. Hence, C is the kernel of the character ϕ : G → F ∗ ℓ / ( F ∗ ℓ ) ∼ = {± } which makes the following diagram commute:Gal( Q /K ) ρ E,ℓ / / / / G ϕ / / det (cid:15) (cid:15) (cid:15) (cid:15) χ ℓ ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ F ∗ ℓ / ( F ∗ ℓ ) ∼ = {± } Gal( K ( ζ ℓ ) /K ) (cid:31) (cid:127) / / F ∗ ℓ ∼ = Aut( µ ℓ ) ❧❧❧❧❧❧❧❧❧❧❧❧❧❧ Q /K ) induced by ϕ is not trivial because there isan element in G with non square determinant since √− ℓ is not in K . ByGalois theory, the kernel of ϕ corresponds to a quadratic extension of K ,which contains √− ℓ by construction. This implies that the extension overwhich E admits an ℓ -isogeny is K ( √− ℓ ).Combining Proposition 3.3 and Theorem 2.2, we have proved the follow-ing result (which is Proposition 1.2 of the Introduction): Proposition 3.4.
Let E be an elliptic curve defined over a number field K , with j ( E ) / ∈ { , } . Let ℓ be a prime number and let q(cid:0) − ℓ (cid:1) ℓ / ∈ K .Suppose that E/K admits an ℓ -isogeny locally at a set of primes with densityone, then E admits an ℓ -isogeny over K ( √− ℓ ) . Moreover, if ℓ = 2 , or ℓ ≡ then E admits an ℓ -isogeny over K . Now let us assume that q(cid:0) − ℓ (cid:1) ℓ belongs to K .As in the previous case, in order to analyse the local-global principle,we study subgroups of SL ( F ℓ ) / {± } which do not fix any point of P ( F ℓ )but whose elements do fix points. The following lemma is a variation on theresult, due to Sutherland, that we stated in the present article as Lemma 2.3,see [Sut12, Lemma 1 and Proposition 2]. This lemma will be the key inunderstanding the case in which q(cid:0) − ℓ (cid:1) ℓ belongs to K . We will denote by S n (resp. A n ) the symmetric (resp. alternating) group on n elements. Lemma 3.5.
Let G be a subgroup of GL ( F ℓ ) whose image H in PGL ( F ℓ ) is contained in SL ( F ℓ ) / {± } . Suppose | P ( F ℓ ) g | > for all g ∈ G but | P ( F ℓ ) G | = 0 . Then ℓ ≡ and one of the followings holds: (1) H is dihedral of order n , where n ∈ Z > is a divisor of ℓ − ; (2) H is isomorphic to one of the following exceptional groups: A , S or A .Proof. No subgroup of GL ( F ) satisfies the hypotheses of the lemma, so weassume ℓ >
2. The orbit-counting lemma yields: | P ( F ℓ ) /H | = 1 | H | X h ∈ H | P ( F ℓ ) h | ≥ | H | ( ℓ + | H | ) > h ∈ H we have | P ( F ℓ ) h | > | P ( F ℓ ) h | = ( ℓ +1) when h isthe identity.If ℓ divides the order of H , then H contains an element h of order ℓ and P ( F ℓ ) /h consists of two orbits, of sizes 1 and ℓ , therefore a fortiori(1 < ) | P ( F ℓ ) /H | ≤
2. But this contradicts the assumption | P ( F ℓ ) H | = 0.Hence ℓ does not divide | H | . 8y Dickson’s classification of subgroups of PGL ( F ℓ ) it follows that H can be either cyclic, or dihedral or isomorphic to one the following groups: S , A , A . To exclude the cyclic case, suppose for the sake of contradictionthat H = h h i . This implies that P ( F ℓ ) h = P ( F ℓ ) H and since | P ( F ℓ ) h | > | P ( F ℓ ) H | = 0. Hence H iseither dihedral, or isomorphic to S , or to A , or to A .By [Sut12, Proposition 2], since H is contained in SL ( F ℓ ) / {± } , thesize of the set of h -orbits of P ( F ℓ ) is even for each h ∈ H . Moreover, as | P ( F ℓ ) h | >
1, all h are diagonalizable on F ℓ . Then, applying the orbit-counting lemma, we have | P ( F ℓ ) /h | = 1ord( h ) X h ′ ∈h h i | P ( F ℓ ) h ′ | == 1ord( h ) ((ord( h ) − ℓ +1) = 2 + ℓ − h ) (1)where h h i denotes the cyclic subgroup of H generated by h . In particular,for elements of order 2 this implies that ℓ ≡ H is a dihedral group of order 2 n . Then there exists h ∈ H of order n , so equation (1) implies that n divides ℓ − Corollary 3.6.
Let G be a subgroup of GL ( F ℓ ) whose image H in PGL ( F ℓ ) is contained in SL ( F ℓ ) / {± } . Suppose | P ( F ℓ ) g | > for all g ∈ G but | P ( F ℓ ) G | = 0 . If H is dihedral of order n , where n ∈ Z > is a divisorof ℓ − , then G is properly contained in the normalizer of a split Cartansubgroup and P ( F ℓ ) /G contains an orbit of size . Let us now focus on case (2) of Lemma 3.5:
Corollary 3.7.
Let G be a subgroup of GL ( F ℓ ) whose image H in PGL ( F ℓ ) is contained in SL ( F ℓ ) / {± } . Suppose | P ( F ℓ ) g | > for all g ∈ G but | P ( F ℓ ) G | = 0 . Then: • if H is isomorphic to A then ℓ ≡ ; • if H is isomorphic to S then ℓ ≡ ; • if H is isomorphic to A then ℓ ≡ .Proof. This is an application of the orbit-counting lemma. In A there areelements of order 2 and 3, and we have ℓ> ( F ) ≃ S . Theequation (1) for elements of order 3 implies that ℓ − ℓ ≡ ℓ ≡ ( F ℓ ).9ince H is contained in SL ( F ℓ ) / {± } , the value of equation (1) has to beeven for every h in H . If H is isomorphic to S , then it contains elements oforder 4 and this implies that ℓ − ℓ ≡ H is isomorphic to A then ℓ>
5: not all matrices inSL ( F ) / {± } ≃ A leave a line stable. There are elements of order 3and 5, then ℓ − ℓ ≡ ℓ ≡ Proposition 3.8.
Let E be an elliptic curve over a number field K of degree d over Q , with j ( E ) / ∈ { , } , and let ℓ be a prime number. Let us suppose q(cid:0) − ℓ (cid:1) ℓ ∈ K and that E/K admits an ℓ -isogeny locally at a set of primeswith density one. Then: (1) if ℓ ≡ the elliptic curve E admits an ℓ -isogeny over K ; (2) if ℓ ≡ the elliptic curve E admits an ℓ -isogeny over a finite ex-tension of K , which can ramify only at primes dividing the conductorof E and ℓ . Moreover, if ℓ ≡− or if ℓ ≥ d +1 , then E admitsan ℓ -isogeny over a quadratic extension of K .Proof. Since q(cid:0) − ℓ (cid:1) ℓ is contained in K , the projective image H of theGalois representation ρ E,ℓ is contained in SL ( F ℓ ) / {± } , as discussed inRemark 2.4, so we can apply Lemma 3.5. This means that if the pair( ℓ, j ( E )) is an exceptional pair then ℓ ≡ H has to be either adihedral group of order 2 n , with n dividing ℓ −
1, or an exceptional subgroup.If the elliptic curve E admits an ℓ -isogeny over a number field L/K thenthere exists a one dimensional Gal( Q /L )-stable subspace of E [ ℓ ]. This sub-space corresponds to a subgroup of the image of the Galois representation.In particular, the extension L of K over which the isogeny is defined canonly ramify at primes where the representation is ramified, that is, only atprimes of K dividing the conductor of E or ℓ .If ℓ ≥ d +1 then H cannot be isomorphic to A , or to S or A ;for a proof of this fact see [Maz77a, p.36]. Analogously, by Corollary 3.7,exceptional images cannot occur if ℓ ≡ − E isconjugated to the normalizer of a split Cartan subgroup which contains theCartan subgroup itself with index 2. By Galois theory, then E admits anisogeny over a quadratic extension of K .Let us now describe all the possibilities that can occur at 2, 3 and 5: Proposition 3.9.
Let K be a number field. There exists no exceptionalpair for K with ℓ = 2 , . If √ belongs to K then there exist exceptionalpairs (5 , j ( E )) for K and, moreover, for such a pair, the projective image of E, (Gal( Q /K )) is, up to conjugation, a dihedral group of order dividing .If √ does not belong to K then there exist no exceptional pairs for K with ℓ = 5 .Proof. As remarked in the proof of Theorem 2.2, for ℓ =2 there exists noexception to the local-global principle. Take ℓ =3. If √− K thenthere exists no exceptional pair since, by Lemma 2.3, the projective imageis a dihedral group of order 2 n with n ∈ Z > odd (dividing 3 − √− K there exists no exceptional pair since, by Lemma 3.5, ℓ ≡ ℓ =5 we have that if √ K then there exists noexceptional pair by Lemma 2.3. Moreover, if √ K then by Lemma 3.5combined with Corollary 3.7, the projective image can only be a dihedralgroup of order dividing 8.We will come back to the study of the local-global principle about 5-isogenies in Section 5.2. In this section we prove statement (1) of the Main Theorem.
Let M be a complete field with respect to a discrete valuation v , which isnormalized, i.e. v ( M ∗ ) = Z . Let O M be its ring of integers, λ the maximalideal of O M and k = O M /λ the residue field. We suppose M of characteristic0, the residue field k finite of characteristic ℓ> e = v ( ℓ ). Let E be anelliptic curve having semi-stable reduction over M and let E be its N´eronmodel over O M . Since M is of characteristic 0, we know that E [ ℓ ]( M ) is an F ℓ -vector space of dimension 2. Let E be the reduction of E modulo λ , then E is a group scheme defined over k whose ℓ -torsion is an F ℓ -vector spacewith dimension strictly lower than 2. Hence, the kernel of the reductionmap, can be either isomorphic to F ℓ (ordinary case) or to the whole E [ ℓ ](supersingular case).Serre, in [Ser72, Proposition 11, Proposition 12 and p.272], describedall possible shapes of the image of I ℓ , the inertia subgroup at ℓ , for thesupersingular case and for the ordinary case: Proposition 4.1 ((Serre, supersingular case)) . Let E be an elliptic curveover M , a complete normalized field with respect to the valuation v , and let e = v ( ℓ ) . Suppose that E has good supersingular reduction at ℓ . Then theimage of I ℓ through the Galois representation ρ E,ℓ : Gal(
M /M ) → GL ( F ℓ ) associated to E is cyclic of order either ( ℓ − /e or ℓ ( ℓ − /e . The two cases depend on the action of the tame inertia.11f the tame inertia acts via powers of the fundamental character of level2, and not 1, it follows that the Newton polygon, with respect to the ellipticcurve, is not broken, and that the tame inertia is given by the e -power ofthe fundamental character of level 2. Hence it has a cyclic image of order( ℓ − /e .If the elliptic curve considered is supersingular, but the tame inertiaacts via powers of the fundamental character of level 1, it follows that therelevant Newton polygon is broken, and there are points in the ℓ -torsionof the corresponding formal group which have valuation with denominatordivisible by ℓ (this follows from [Ser72, p.272]). So the image of inertia hasorder ℓ ( ℓ − /e .In the ordinary case, the following proposition holds: Proposition 4.2 ((Serre, ordinary case)) . Let E be an elliptic curve over M , a complete normalized field with respect to the valuation v , and let e = v ( ℓ ) . Suppose that E has semistable ordinary reduction at ℓ . Then theimage of I ℓ through the Galois representation ρ E,ℓ : Gal(
M /M ) → GL ( F ℓ ) associated to E is cyclic of order either ( ℓ − /e or ℓ ( ℓ − /e , and it can berepresented, after the choice of an appropriate basis, respectively as (cid:18) ∗
00 1 (cid:19) or (cid:18) ∗ ⋆ (cid:19) , with ∗ ∈ F ∗ ℓ and ⋆ ∈ F ℓ . Let E λ be an elliptic curve defined over a complete field M with maximalideal λ , residual characteristic ℓ , and let d ′ = j ( E ) , λ, j ( E ) ≡ λ, ℓ ≥
53 if j ( E ) ≡ λ, ℓ ≥ , j ( E ) ≡ λ, ℓ = 3 ,
12 if j ( E ) ≡ λ, ℓ = 2 . (2)Then E λ , or a quadratic twist, has semistable reduction over a finite ex-tension of M with degree d ′ , see for instance [BK75, pp.33 −
52] or [Kra90,Proposition 1 and Th´eor`eme 1].We now give the main result of this article:
Theorem 4.3.
Let ( ℓ, j ( E )) be an exceptional pair for the number field K of degree d over Q , such that q(cid:0) − ℓ (cid:1) ℓ / ∈ K and j ( E ) / ∈ { , } . Then ℓ ≡ and ≤ ℓ ≤ d +1 . roof. Since the pair ( ℓ, j ( E )) is exceptional, E admits an ℓ -isogeny locallyat a set of primes with density one, ℓ ≡ ℓ -isogeny over L = K ( √− ℓ ). In particular, ℓ = 2 , ℓ ≥ K λ be the completion of K at λ , a prime above ℓ , and let M bethe smallest extension of K λ over which E λ := E ⊗ K λ acquires semi-stablereduction. After replacing E by a quadratic twist if necessary, we can assumethat the extension M/K λ has degree less than or equal to 3, according to(2), since ℓ ≥
7. Let E be the reduction of E λ ′ := E ⊗ M modulo λ ′ , for λ ′ the prime above λ .We now look at the image of the inertia subgroup under the Galoisrepresentation associated to the ℓ -torsion of E . The Galois representationhas image of order dividing ( ℓ − by Remark 3.2.Assume that the reduction E is supersingular. The inertia has imageisomorphic to a cyclic group of order ( ℓ − /m or ℓ ( ℓ − /m , where m is lessthan or equal to 3 d , according to the degree of the extension needed to havesemi-stable reduction. In the first case, i.e. when the image of the inertiahas order ( ℓ − /m , it is isomorphic to a non-split torus in GL ( F ℓ ) and isalso contained in the normalizer of a split Cartan, as stated in Lemma 2.3.This is impossible unless ( ℓ − /m | ( ℓ − . This means that ( ℓ +1) /m | ( ℓ − ℓ +1) /m |
2, hence ℓ ≤ m − ℓ ( ℓ − /m divides ( ℓ − ,therefore ℓ ≤ m ≤ d . The pair ( ℓ, j ( E )) is exceptional, so ℓ ≡ ℓ ≥
7, hence, 7 ≤ ℓ ≤ m − ≤ d − ℓ > m −
1, then E is not supersingular, so it isordinary, since E λ is semistable over M . By Proposition 3.3, then the ellip-tic curve E admits two ℓ -isogenies over L = K ( √− ℓ ), which are conjugateover L . By Lemma 2.3, the image G of Gal( Q /K ) acting on E [ ℓ ]( K ) is asubgroup of the normalizer N of a split Cartan subgroup C . From Proposi-tion 3.3, we know that N/C ≃ Gal(
L/K ) = { } , so the image of an inertiasubgroup I λ at the place λ of K is a subgroup of G whose image in N/C isnon-trivial. On the other hand, Proposition 4.2 shows that, if ( ℓ − /m > I λ contains a cyclic subgroup of order larger than or equal to 3 (foranother argument see [Maz77b, p.118]). It follows that I λ contains a non-trivial Cartan subgroup (of shape (cid:26)(cid:18) a b (cid:19) : a, b ∈ Γ (cid:27) for a certain nontrivial subgroup Γ of F ℓ ), even after restriction of the scalars to Gal( M /M ).This is a contradiction with Proposition 4.2, as the latter says that the re-striction of I λ to Gal( M /M ) is a semi-Cartan subgroup (or a Borel). Hence,( ℓ − /m ≤ ℓ ≡ ≤ ℓ ≤ m +1 ≤ d +1. Remark . It is clear that Theorem 4.3 implies the result obtained bySutherland in the case K = Q . 13he previous Theorem, combined with Remark 2.4, implies point (1) ofthe Introduction’s Main Theorem, namely: Corollary 4.5.
Let ( ℓ, j ( E )) be an exceptional pair for the number field K of degree d over Q and discriminant ∆ . Assume j ( E ) / ∈ { , } . Then ℓ ≤ max {| ∆ | , d +1 } . Proof.
If ( ℓ, j ( E )) is an exceptional pair for the number field K then we dis-tinguish two cases according to the projective image being contained or notin SL ( F ℓ ) / {± } . This corresponds to a condition about q(cid:0) − ℓ (cid:1) ℓ belongingto K or not. If q(cid:0) − ℓ (cid:1) ℓ / ∈ K we can apply Theorem 4.3 and conclude that7 ≤ ℓ ≤ d +1. While, if q(cid:0) − ℓ (cid:1) ℓ ∈ K , then ℓ divides | ∆ | . Given a number field K of degree d over Q and discriminant ∆, the local-global principles for ℓ -isogenies holds whenever ℓ > ℓ k := max {| ∆ | , d +1 } or ℓ = 2 or 3 by Corollary 4.5 and Proposition 3.9. In this section we analysewhat happens for primes smaller than the bound obtained. In particular wewill prove that the local-global principle about ℓ -isogenies for elliptic curvesover number fields admits only a finite number of exceptions if ℓ >
7. Wewill also study the behaviour of the local-global principle at 5 and 7.Let K be a number field and let C/K be a projective smooth curvedefined over K and of genus g . Our arguments will rely on the classicaltrichotomy between curves of genus 0, 1 and higher. When the genus is 0,the curve is isomorphic to P K over an algebraic closure of K and therefore C ( K ), the set of K -rational points, is either empty or infinite. If the genusof C is 1 and C ( K ) contains at least one point over K then C/K is an ellipticcurve over K and the Mordell-Weil theorem shows that C ( K ) is a finitelygenerated abelian group: C ( K ) ∼ = T ⊕ Z r , where T is the torsion subgroupand r is a non-negative integer called the rank of the elliptic curve, whereasif g ≥
2, Faltings Theorem states that the set of K -rational points is finite.In this section we will recall some theory of modular curves.Let ℓ ≥ Z [ ζ ℓ ] be the subring of C generatedby a root of unity of order ℓ . The modular curve X ( ℓ ) is the compactifiedfine moduli space which classify pairs ( E, α ), where E is a generalized ellipticcurve over a scheme S over Spec( Z [1 /ℓ, ζ ℓ ]) and α : ( Z /ℓ Z ) S ∼ → E [ ℓ ] is anisomorphism of group schemes over S which is a full level ℓ -structure, up toisomorphism of pairs, i.e. isomorphisms of elliptic curves that preserve thelevel structure. A full level ℓ -structure on a generalized elliptic curve E over14 is a pair of points ( P , P ), satisfying P , P ∈ E [ ℓ ] and e ℓ ( P , P ) = ζ ℓ where e ℓ is the Weil pairing on E [ ℓ ]. Let us recall that a full level ℓ -structureinduces a symplectic pairing on ( Z /ℓ Z ) via h (1 , , (0 , i = ζ ℓ . For moredetails, see [KM85] or [Gro90].Let G be a subgroup of GL ( Z /ℓ Z ). We will denote as X G ( ℓ ) := G \ X ( ℓ )the quotient of the modular curve X ( ℓ ) by the action of G on the full level ℓ -structure. The modular curve X G ( ℓ ) has a geometrically irreducible modelover Q ( ζ ℓ ) det( G ) , the subfield of Q ( ζ ℓ ) invariant under the action of det( G ),see [Maz77b, pp.115 − . . G is the Borel subgroup, X G ( ℓ ) is the modular curve X ( ℓ ) over Q . This modular curve parametrizes elliptic curves with a cyclic ℓ -isogeny, that is, pairs ( E, C ), where E is a generalized elliptic curve and C is the kernel of a cyclic ℓ -isogeny, up to isomorphism.If G is a split Cartan subgroup (respectively, the normalizer of a splitCartan subgroup) we will denote the modular curve X G ( ℓ ) := X sp . Car ( ℓ )(respectively, X split ( ℓ )). The curve X sp . Car ( ℓ ) (respectively, X split ( ℓ )) pa-rametrizes elliptic curves endowed with an ordered (respectively, unordered)pair of independent cyclic ℓ -isogenies.Following [Maz77b], we will denote as X A ( ℓ ) (respectively X S ( ℓ ), X A ( ℓ ))the modular curves obtained taking as G ⊂ GL ( Z /ℓ Z ) the inverse imageof A ⊂ PGL ( Z /ℓ Z ) (respectively S , A ⊂ PGL ( Z /ℓ Z )). Let us remarkthat exceptional projective images A , S and A can occur only for particu-lar values of ℓ , see [Ser72, section 2 .
5, 2 . X A ( ℓ ) and X A ( ℓ ) have geometrically irreducible models over the quadratic subfield of Q ( ζ ℓ ). The same holds for X S ( ℓ ) if ℓ
6≡ ± Q . Remark . Let
E/K be an elliptic curve that arises in an exceptional pair( ℓ, j ( E )) for the number field K . Let us suppose that the projective im-age of ρ E,ℓ is dihedral. Hence, (
E, ρ
E,ℓ ) corresponds to a K -rational pointin X split ( ℓ ) by Lemma 2.3 and Corollary 3.6. Moreover, E [ ℓ ]( K ) containstwo conjugate lines L and L over L/K , where
L/K is quadratic (Propo-sitions 3.4 and 3.8). These lines correspond to the isogenies α : E → E/L and β : E → E/L defined over L . Hence, they give a pair of L -rationalpoints (taking respectively αβ ∨ and βα ∨ as isogeny structure) on X ( ℓ )which are conjugate by the Fricke involution w ℓ ; for a definition see [Par05,section 2]. Let us recall that there exists an isomorphism defined over Q between X ( ℓ ) and X sp . Car ( ℓ ). Remark . If ( ℓ, j ( E )) is an exceptional pair for the number field K and q(cid:0) − ℓ (cid:1) ℓ / ∈ K then the prime ℓ is congruent to 3 mod 4 and hence to 7 or11 mod 12, while if q(cid:0) − ℓ (cid:1) ℓ ∈ K then the prime ℓ is congruent to 1 mod 4and hence to 1 or 5 mod 12. 15 .1 The case ≤ ℓ ≤ ℓ K Theorem 5.3.
Let K be a number field. If ℓ is a prime greater than , thenthe number of exceptional pairs ( ℓ, j ( E )) for K is finite.Proof. Given an exceptional pair ( ℓ, j ( E )) for the number field K it cor-responds to a K -rational point on one of the following modular curves: X split ( ℓ ), X A ( ℓ ), X S ( ℓ ) or X A ( ℓ ), by Lemmas 2.3 and 3.5. Let us analyseeach possible case.Let us recall that the genus of X split ( ℓ ) is given by the following formula,see [Maz77b, p. 117]: g ( X split ( ℓ )) = 124 (cid:18) ℓ − ℓ + 11 − (cid:18) − ℓ (cid:19)(cid:19) . Hence, if ℓ ≡ g ( X split ( ℓ )) = ( ℓ − ℓ + 7). Otherwise,if ℓ ≡ g ( X split ( ℓ )) = ( ℓ − ℓ + 15). Therefore, themodular curve X split ( ℓ ) has genus greater than or equal to 2 for ℓ ≥
11, andhas only finitely many K -rational points by Faltings Theorem.Let us now study the modular curves X A ( ℓ ), X S ( ℓ ) and X A ( ℓ ). Thegenus of these modular curves is given by the following formulae, see [CH05,section 2]: g ( X A ( ℓ )) = 1288 ( ℓ − ℓ − ℓ + 294 + 18 ǫ + 32 ǫ ) g ( X S ( ℓ )) = 1576 ( ℓ − ℓ − ℓ + 582 + 54 ǫ + 32 ǫ ) g ( X A ( ℓ )) = 11440 ( ℓ − ℓ − ℓ + 1446 + 90 ǫ + 80 ǫ )where ǫ is equal to 1 if ℓ ≡ − ℓ ≡ ǫ isequal to 1 if ℓ ≡ − ℓ ≡ − ℓ , see [Ser72, section2 .
5, 2 . ℓ ,as already noticed in [CH05, p.3072]. By Corollary 3.7, if an exceptional pairhas projective image isomorphic to A then ℓ ≡ X A ( ℓ ) is greater than 2 for all ℓ ≥
13. Similarly, for projectiveimage isomorphic to S or to A the genus of the respective modular curvesis larger than 2 for primes satisfying the appropriate congruence. ℓ = 5 Now we will study the local-global principle for 5-isogenies. In order to doso it will be relevant to recall the structure of X (5) at the cusps.The modular interpretation of X (5)( Q ) associates to each cusp a N´eronpolygon P with 5 sides. The N´eron polygon is endowed with the structure16f a generalized elliptic curve enhanced with a basis of P [5] ∼ = µ × Z / Z ,where µ is the set of 5-th roots of unity, up to automorphisms of P :( {± } × µ ) × P [5] → P [5] (cid:18)(cid:18) ǫ α ǫ (cid:19) , (cid:18) wj (cid:19)(cid:19) (cid:18) w ǫ α j ǫj (cid:19) where ǫ ∈{± } and α, w ∈ µ . The set of cusps of X (5)( Q ) is a Galoisset with an action of GL ( Z / Z ). The modular interpretation of X G (5)associates to each cusp an orbit of the enhanced N´eron polygon under theaction of G .The local-global principle for 5-isogenies is related to V , the Klein 4-group. Let us recall that there is a unique non-trivial 2-dimensional irre-ducible projective representation τ of V in PGL ( F ) and, up to conjuga-tion, this representation is given by the image in PGL ( F ) of the set: (cid:26)(cid:18) (cid:19) , (cid:18) −
11 0 (cid:19) , (cid:18) − (cid:19) , (cid:18) (cid:19)(cid:27) . (3)For a prime ℓ ≥
5, we denote as X V ( ℓ ) the modular curves X G ( ℓ ) obtainedby taking as G ⊂ GL ( Z /ℓ Z ) the inverse image of V ⊂ PGL ( Z /ℓ Z ). Proposition 5.4.
Over
Spec( Q ( √ the modular curve X V (5) is isomor-phic to P .Proof. The genus of X (5) over Q ( ζ ) is 0. The field of constants of X V (5) is Q ( ζ ) det( G ) where G is the inverse image of V in GL ( F ). Since V ⊂ SL ( F ) / {± } and F ∗ ⊂ G , then det( G ) = ( F ∗ ) . This means that X V (5) is geometricallyirreducible over Q ( √
5) and its genus is 0.The set of cusps of X (5)( Q ) is in 1 − F -vector spaces between F and µ × F by the action of {± } × µ . To show that over Spec( Q ( √ X V (5) is isomorphic to P it is enough to show that the set of Q ( √ X V (5) is non-empty.Let φ ζ : F ∼ → µ × F be the isomorphism given by φ ζ ((1 , ζ , φ ζ ((0 , , φ ζ under theaction of {± } × µ is the following set: { (( ζ , , (1 , , (( ζ − , , (1 , − , (( ζ , , ( ζ , , (( ζ − , , ( ζ − , − , (( ζ , , ( ζ , , (( ζ − , , ( ζ − , − , (( ζ , , ( ζ − , , (( ζ − , , ( ζ , − , (( ζ , , ( ζ − , , (( ζ − , , ( ζ , − } . On the set of cusps we have a Galois action by Gal( Q ( ζ ) / Q ). In partic-ular, acting with − ∈ Gal( Q ( ζ ) / Q ) on the cusp (( ζ , , (1 , ζ − , , (1 , X (5)( Q ).17owever, the action of − ∈ Gal( Q ( ζ ) / Q ) on the class of the cusp(( ζ , , (1 , G : in fact G , the inverseimage of V in GL ( F ), is the group (cid:26)(cid:18) x ± x (cid:19) , (cid:18) ± xx (cid:19) : x ∈ F ∗ (cid:27) , and under the action of (cid:18) − (cid:19) ∈ G the cusp (( ζ − , , (1 , X (5)( Q )is mapped to (( ζ , , (1 , ζ , , (1 , X V (5)( Q ) is stable underGal( Q ( ζ ) / Q ( √ X V (5)( Q ( √ Remark . The argument given above shows that X V (5)( Q ( √ Q ( √ Q ( √ X V (5). Over Q , the mod-5 Galois image of the elliptic curve with LMFDBlabel 608 . e1 : E : y = x − x + 4848is equal to the normalizer of the split Cartan in GL ( F ): up to conjugacy,the normalizer of the split Cartan is the only subgroup of GL ( F ) withorder 32 and [ Q ( E [5]) : Q ] = 32. Changing the field of definition of E to Q ( √
5) reduces the Galois image to the index-2 subgroup of the normalizerof the split Cartan with square determinants. The image of this subgroupin PGL ( F ) is isomorphic to V , therefore j ( E ) corresponds to a Q ( √ X V (5). Through the LMFDB database, see [LMF13a]and in particular the search engine [LMF13b], it is possible to find lotsof examples of elliptic curves whose mod-5 Galois image is equal to thenormalizer of the split Cartan in GL ( F ), such as the elliptic curves withLMFDB labels 121 . b2 and 1216 . h1. Corollary 5.6.
There exist infinitely many exceptional pairs (5 , j ( E )) forthe number field K if and only if √ belongs to K .Proof. By Proposition 3.9, there is an exceptional pair (5 , j ( E )) for thenumber field K only if √ K .If (5 , j ( E )) is an exceptional pair for the number field K then the pro-jective image of the Galois representation associated to the elliptic curve E over K is a dihedral group of order dividing 8 (Lemma 3.5 combined withCorollary 3.7). If √ K , each of the infinitely many (Proposi-tion 5.4) non-cuspidal points on X V (5)( K ) corresponds to the isomorphismclass of an elliptic curve E whose Galois image in PGL 2( F ) is isomorphic to V . Every group G ⊂ GL ( F ) with image V in PGL ( F ) has the property see [LMF13b], Cremona label 608b1 F but G does not, for example,no line in F is fixed by both the second and third matrices in the set (3).Therefore all the pairs (5 , j ( E )) are exceptional. Remark . In [BC13], it is given an explicit formulafor the map from X V (5) to X (1) and, hence, an explicit parametrization of the j -invariantsof the exceptional pairs (5 , j ( E )). ℓ = 7 The local-global principle for 7-isogenies leads us to a dichotomy between afinite and an infinite number of counterexamples according to the rank of aspecific elliptic curve that we call the Elkies-Sutherland curve:
Proposition 5.8. If ℓ = 7 then the number of exceptional pairs (7 , j ( E )) for a number field K , is finite or infinite, depending whether the rank of theelliptic curve E ′ : y = x − x + 33614 is zero or non-zero, respectively.Proof. If √− ∈ K then by Lemma 3.5 there is no exceptional pair. Let ussuppose that √− K . As shown by Sutherland in [Sut12, Section3] and explained in Remark 5.1, the modular curve of interest is the twistof X (49) by Gal( K ( √− /K ) with respect to w , the Fricke involution on X (49). As explained in [Sut12, Section 3], computations of Elkies implythat this modular curve is isomorphic to E ′ , thus each K ( √− E ′ gives rise to an exceptional pair (7 , j ( E )). Explicitly, if the K ( √− E ′ has coordinates ( u, v ), let t = (7 u − v +343) / v ,then the j -invariant of the isomorphism class of elliptic curves which areexceptional for the local-global principle for 7-isogenies is equal to − ( t − ( t − t + t − ( t + t + 2) ( t − t + 2 t + 3 t + 1) ( t − t − t + 1) . Hence, if the rank of E ′ over K is positive there are infinitely many coun-terexamples to the local-global principle about 7-isogenies, while if the rankis 0 there are only finitely many. Remark . As shown by Sutherland in [Sut12, Section 3], over Q ( i ) thecurve E has positive rank, so there are infinitely many counterexamples overthis field to the local-global principle for 7-isogenies.The proof of our Main Theorem is now complete.19 Complex multiplication
Sutherland in [Sut12] proved that an exceptional pair ( ℓ, j ( E )) over Q can-not have complex multiplication: for ℓ > ℓ = 7 we refer to the direct computations in [Sut12,Section 3]. Here we study the same problem for K a number field. Lemma 6.1.
Let K be a number field of degree d over Q and let E/K be anelliptic curve over K with j ( E ) / ∈ { , } . Let ( ℓ, j ( E )) be an exceptionalpair for K . If ℓ > d +1 then E does not have complex multiplication.Proof. Assume that E has complex multiplication by a quadratic order O .This means that the Galois representation ρ E,ℓ has image included in a Borel,when ℓ ramifies in O , or is projectively dihedral (split or nonsplit, accordingto ℓ being split or inert in O ), see [Ser66, Th´eor`eme 5]. The Borel caseis clearly not possible. By Proposition 3.4, there is an ℓ -isogenous ellipticcurve E ′ that is defined over a quadratic extension L of K , but not over K ,since we are in an exceptional case. Since E ′ is isogenous to E over L , italso must have complex multiplication by an order O ′ . Since E and E ′ are ℓ -isogenous, by [Cox89, Theorem 7 . h ( O ′ ) and h ( O ) satisfies h ( O ′ ) h ( O ) = 1[ O ∗ : O ′∗ ] (cid:18) ℓ − (cid:18) disc( O ) ℓ (cid:19)(cid:19) ≥ ( ℓ − , since j ( E ) / ∈ { , } and therefore [ O ∗ : O ′∗ ] = 1. In particular, if h ( O ′ ) = h ( O ) we have a contradiction. Hence assume h ( O ′ ) > h ( O ). Since E ′ is defined over L and E is defined over K , we know that Q ( j E ) ⊆ K and Q ( j E ′ ) ⊆ L . Then the ratio of the class numbers h ( O ′ ) and h ( O )satisfies: h ( O ′ ) h ( O ) ≤ [ L : Q ] = 2 d. Hence, if ℓ > d + 1, we have a contradiction between the lower and theupper bound, so E cannot have complex multiplication. Acknowledgements
I would like to thank my advisor Pierre Parent, for his advice, his guid-ance and patience throughout the many readings and corrections of thisarticle. I also am very grateful to my advisor Bas Edixhoven, for all thefruitful discussions about this topic. I would also like to thank the refereefor the thorough, constructive and helpful comments and suggestions on themanuscript. 20 eferences [BC13] Barinder Singh Banwait and John Cremona. Tetrahedral el-liptic curves and the local-to-global principle for isogenies. http://arxiv.org/pdf/1306.6818v2.pdf , 2013.[BK75] Bryan J. Birch and Willem Kuyk.
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E-mail address: s. anni@ warwick. ac. uks. anni@ warwick. ac. uk