Journal of Number Theory NT2036 journal of number theory 62, 144162 (1997) article no. NT972036 Torsion Subgroups of Elliptic Curves over Quadratic Extensions Soonhak Kwon Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218 Communicated by A. Hildebrand Received November 13, 1995; revised January 19, 1996 Let E be an elliptic curve over Q. Observing certain relations between the torsion subgroups of E and ED , the D-quadratic twist of E, we prove that the torsion View metadata, citation and similar papers at core.ac.uk brought to you by CORE subgroups of E is stable for all but finitely many quadratic extensions. Moreover, using the result of K. Ono, we classify the torsion subgroup of E over all quadratic provided by Elsevier - Publisher Connector extensions when E is of the form E: y2=x(x+M)(x+N), where M and N are integers. In the special case when torsion subgroup of E over Q is isomorphic to ZÂ2ZÄZÂ8Z, we prove that the torsion subgroup of E is always stable under quadratic extensions. 1997 Academic Press 1. INTRODUCTION Let E be an elliptic curve defined over Q. Let T(E, Q) be the torsion subgroup of the group of rational points of E over Q. Similarly, we define T(E, Q(- D)) for any quadratic extension Q(- D) over Q where D is a square free integer. T(E, Q) is a subgroup of T(E, Q(- D)). Letting E: y2=x3+Ax+B, where A and B are integers, we define the 2 3 D-quadratic twist of E, denoted by, ED : Dy =x +Ax+B. Sometimes, 2 3 2 3 we use the equation ED : y =x +D Ax+D B which is isomorphic to previous one by the map which sends (x, y)to(xÂD, yÂD2). Let _ be a generator of Gal(Q(- D)ÂQ). In this paper we are going to prove the following theorems which explain the properties of torsion subgroups of elliptic curves under quadratic extensions. Theorem 1. Let E: y2=x(x+M)(x+N), where M and N are integers, be an elliptic curve over Q. Assume (M, N) is a square free integer. Then for all quadratic extensions Q(- D) over Q, where D is a square free integer, the torsion subgroup of E over Q(- D) can be classified as follows. 144 0022-314XÂ97 25.00 Copyright 1997 by Academic Press All rights of reproduction in any form reserved. File: 641J 203601 . By:CV . Date:22:01:97 . Time:12:42 LOP8M. V8.0. Page 01:01 Codes: 3731 Signs: 2014 . Length: 50 pic 3 pts, 212 mm TORSION SUBGROUPS OF ELLIPTIC CURVES 145 (i) If T(E, Q)=ZÂ2ZÄZÂ8Z, then T(E, Q(- D))=ZÂ2ZÄZÂ8Z for all D. (ii) If T(E, Q)=ZÂ2ZÄZÂ6Z, then by defining a set S=[M, N, &M,&M+N,&N,&N+M], it has a following classification.(a)if none of the integers in S are squares, then T(E, Q(- D))=ZÂ2ZÄZÂ6Z for all D. (b) if there is a square in S, then it is the only square in the set S and it determines the unique D such that T(E, Q(- D))=ZÂ4ZÄZÂ6Z and T(E, Q(- D$))=ZÂ2ZÄZÂ6Z for all other D$ different from D. (iii) If T(E, Q)=ZÂ2ZÄZÂ4Z, then after linear change of coordinates we assume M and N are squares. In this case,(a)T(E,Q(-D))=ZÂ4ZÄ ZÂ4Z when D=&1, M=s2, N=t2 and s2&t2=\r2 for some integers s, t and r.(b)T(E,Q(-D))=ZÂ2ZÄZÂ8Z when M=u4, N=v4, where u2+v2 is not a square and D is the square free part of the factorization of u2+v2 or when M=u4, N=D2v4 (resp. M=D2v4, N=u4) where u, v and w are integers for which u2+Dv2=w2.(c)T(E,Q(-D))=ZÂ2ZÄZÂ4Z for all other cases. (iv) If T(E, Q)=ZÂ2ZÄZÂ2Z, then T(E, Q(- D)) may take any of ZÂ2ZÄZÂ2Z, ZÂ2ZÄZÂ4Z, ZÂ2ZÄZÂ6Z, ZÂ2ZÄZÂ8Z, ZÂ4ZÄZÂ6Z, and this can be determined using a standard isomorphism between T(E, Q(- D)) and T(ED , Q(- D)). Theorem 2. Let E: y2=x(x+M)(x+N), where M and N are integers, be an elliptic curve over Q. Then, (i) If T(E, Q)=ZÂ2ZÄZÂ8Z, then T(ED , Q)=ZÂ2ZÄZÂ2Z for all square free integers D. (ii) If T(E, Q)=ZÂ2ZÄZÂ6Z, then T(ED , Q)=ZÂ2ZÄZÂ2Z for all square free integers D. (iii) If T(E, Q)=ZÂ2ZÄZÂ4Z, then after linear change of coordinates we assume M=s2, N=t2 for some integers s and t.(a)If D=&1 and s2&t2 2 =\r for some integer r, then T(ED , Q)=ZÂ2ZÄZÂ4Z.(b)If M, N and D do not satisfy the condition of part (a), then T(ED , Q)=ZÂ2ZÄZÂ2Z. (iv) If T(E, Q)=ZÂ2ZÄZÂ2Z, then T(ED , Q)=ZÂ2ZÄZÂ4Z, ZÂ2ZÄZÂ6Z, ZÂ2ZÄZÂ8Z for only finitely many D and T(ED , Q)= ZÂ2ZÄZÂ2Z for almost all D. 2. QUADRATIC TWISTS AND TORSION SUBGROUPS For a given elliptic curve E over Q, the group of rational points has many interesting properties related with quadratic extension Q(- D). For File: 641J 203602 . By:CV . Date:22:01:97 . Time:12:42 LOP8M. V8.0. Page 01:01 Codes: 3131 Signs: 1993 . Length: 45 pic 0 pts, 190 mm 146 SOONHAK KWON example, one can easily show that the rank of E over Q(- D) is the sum of the ranks of E and ED over Q. In the case of torsion subgroups, we have the following simple proposition. Proposition . 1 The map h: T(E, Q(- D))ÂT(E, Q) Ä T(ED , Q) defined by h(P ):=P&_P is an injective map of abelian groups. Proof. Since P&_P is in the kernel of the trace map Tr: Q [ Q+_Q, it is of the form (x0 , y0 - D) where x0 and y0 are in Q. P&_P in T(ED , Q) is understood as (x0 , y0)inT(ED,Q). The map is well defined because T(E, Q) is in the kernel of h.IfP&_P=Q&_Q, then _ fixes the point P&Q. Therefore P&Q is in T(E, Q), which implies that h is injective. K Corollary 1. Let E be an elliptic curve defined over Q. Then, its torsion subgroup over Q(- D) is bounded for any quadratic extension Q(- D). More precisely, for all square free D, |T(E, Q(- D))|E162. Proof. By Mazur's classification of torsion subgroups of elliptic curves over Q, the largest possible torsion subgroup of E over Q is of order 16. By Proposition 1, |T(E, Q(- D))|E|T(E, Q)|}|T(ED, Q)|E16 } 16. K Note that the bound 162 can be greatly improved once we prove Theorem 1 and 2. Above corollary is closely related with the uniform boundness conjecture which says that, for a given number field K, there is a constant B depending only on the extension degree of K over Q such that the order of torsion subgroup of any elliptic curve over K is bounded by B. This difficult conjecture has been proved recently by S. Kamienny [3] for quadratic case and by L. Merel [10] for general cases. Corollary 2. Let E be an elliptic curve defined over Q. Then, for all square free integers D with only finitely many exceptions, T(E, Q(- D))=T(E, Q). i.e. The torsion subgroups are stable under almost all quadratic extensions. File: 641J 203603 . By:CV . Date:22:01:97 . Time:12:42 LOP8M. V8.0. Page 01:01 Codes: 2669 Signs: 1743 . Length: 45 pic 0 pts, 190 mm TORSION SUBGROUPS OF ELLIPTIC CURVES 147 Proof. By Proposition 1, T(E, Q(- D))ÂT(E, Q) is isomorphic to a subgroup of T(ED , Q). Therefore since any point in T(ED , Q) has an possible order 1, 2, ..., 10 or 12 by Mazur's classification, T(E, Q) % T(E, Q(- D)) if and only if there exist P # T(E, Q), Q # T(E, Q), m#[2, 3, ..., 10, 12] such that mQ=P, Q(Q)=Q(x, y)=Q(-D) where Q=(x, y). Note that there are only finitely many P # T(E, Q) and m # [2, 3, ..., 10, 12]. Thus, there are only finitely many Q # T(E, Q ) such that mQ=P. Among them, only finitely many Q such that Q(Q)=Q(-D) for some D # Z. Excluding those D's which appear as Q(Q)=Q(-D), we get T(E, Q)=T(E, Q(-D)) for all other D's. K It should be noted that above corollary can be deduced from Serre's deep theorem (see [14]), which says that Galois representation induced by l-torsion points is surjective for all but finitely many primes l. But the proof given above provides a concrete upper bound on the number of quadratic fields for which the torsion subgroup is not stable because, given m and P, there are exactly m2 solutions of Q for the equation mQ=P. It is not difficult to see that we only need primes l(l=2, 3, 5 and 7), instead of m # [2, 3, ..., 10, 12], to find a point Q in a quadratic extension such that lQ=P where P is in T(E, Q). By solving the equation lQ=P for every possible l and P, we can determine those quadratic fields for which the torsion subgroup is not stable.