Article Contentsp. 121p. 122p. 123p. 124p. 125p. 126p. 127p. 128p. 129p. 130p. 131p. 132p. 133p. 134p. 135Issue Table of ContentsTransactions of the American Mathematical Society, Vol. 264, No. 1 (Mar., 1981), pp. 1-254Volume InformationFront MatterThe Genus of a Map [pp. 1-28]Orientation-Reversing Morse-Smale Diffeomorphisms on the Torus [pp. 29-37]Schur Products of Operators and the Essential Numerical Range [pp. 39-47]Some General Theorems on the Cohomology of Classifying Spaces of Compact Lie Groups [pp. 49-58]On Spectral Theory and Convexity [pp. 59-75]Submonotone Subdifferentials of Lipschitz Functions [pp. 77-89]Invariance of Solutions to Invariant Nonparametric Variational Problems [pp. 91-111]A Representation-Theoretic Criterion for Local Solvability of Left Invariant Differential Operators on Nilpotent Lie Groups [pp. 113-120]Arithmetic of Elliptic Curves Upon Quadratic Extension [pp. 121-135]The Cohomology Algebras of Finite Dimensional Hopf Algebras [pp. 137-150]Homotopy Groups of the Space of Self-Homotopy-Equivalences [pp. 151-163]Uniqueness of Product and Coproduct Decompositions in Rational Homotopy Theory [pp. 165-180]Which Curves Over $Z$ have Points with Coordinates in a Discrete Ordered Ring? [pp. 181-189]Quasi-Symmetric Embeddings in Euclidean Spaces [pp. 191-204]Characterization of Some Zero-Dimensional Separable Metric Spaces [pp. 205-215]Some Countability Conditions on Commutative Ring Extensions [pp. 217-234]Localizable Analytically Uniform Spaces and the Fundamental Principle [pp. 235-250]Class Groups of Cyclic Groups of Square Free Order [pp. 251-254]Back MatterArithmetic of Elliptic Curves Upon Quadratic ExtensionAuthor(s): Kenneth KramerSource: Transactions of the American Mathematical Society, Vol. 264, No. 1 (Mar., 1981), pp.121-135Published by: American Mathematical SocietyStable URL: http://www.jstor.org/stable/1998414Accessed: 01/09/2009 14:50Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available athttp://www.jstor.org/page/info/about/policies/terms.jsp. 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For more information about JSTOR, please contact [email protected] Mathematical Society is collaborating with JSTOR to digitize, preserve and extend access toTransactions of the American Mathematical Society.http://www.jstor.orgTRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 264, Number 1, March 1981 ARITHMETIC OF ELLIPTIC CURVES UPON QUADRATIC EXTENSION BY KENNETH KRAMER1 ABSTRACT. This paper is a study of variations in the rank of the Mordell-Weil group of an elliptic curve E defined over a number field F as one passes to quadratic extensions K of F. Let S(K) be the Selmer group for multiplication by 2 on E(K). In analogy with genus theory, we describe S(K) in terms of various objects defined over F and the local norm indices iv, = dimF2E(FV)/Norm{E(K14)1 for each completion FV, of F. In particular we show that dim S(K) + dim E(K)2 has the same parity as E:it. We compute iv, when E has good or multiplicative reduction modulo v. Assuming that the 2-primary component of the Tate-Shafare- vitch group 11I(K) is finite, as conjectured, we obtain the parity of rank E(K). For semistable elliptic curves defined over Q and parametrized by modular functions our parity results agree with those predicted analytically by the conjectures of Birch and Swinnerton-Dyer. 1. Introduction. Let E be an elliptic curve defined over a number field F. Our motivating question is this: What can be said about variations in the rank of the Mordell-Weil group E(K) over quadratic extensions K = F(d '/2)? Let E(d) denote the twist of E which becomes isomorphic to E over K but not over F. Concretely, if we choose for E a model over F of the form y2 = f(x) then a model for E(d) is given by dy2 = f(x). If a denotes the generator of Gal(K/F), then E(F) can be identified with the (+1)-eigenspace and E(d)(F) with the (-l)-eigenspace of a acting on E(K). It follows that rank E(K) = rank E(F) + rank E (d)(F). An equiv- alent question therefore is to describe changes in the rank of E(d)(F) as d varies. For certain specific curves defined over Q this question has been discussed for example in [1], [11]. Let N: E(K) -* E(F) be the norm mapping defined naively by N(P) = P + P?. Our starting point is to determine the dimension (as a vector space over F2) of the cokernel of its local counterpart Nw: E(K,) -* E(F) for each completion K, of K. The results depend of course on the ramification in K, over F, and the type of reduction of E. We restrict our attention to semistable (i.e., good or multiplicative) reduction and, in case of residue characteristic 2, an unramified ground field F,. These local calculations are of interest in themselves, and may be read indepen- dently. The situation for cases of additive reduction seems to be more complicated; Received by the editors November 28, 1979; preliminary report presented to the Society April 20, 1979 under the title Elliptic Curves Over Quadratic Fields. AMS (MOS) subject classifications (1970). Primary 14G25, 14K15, 14G20, 1OB10. Key words and phrases. Elliptic curve, quadratic extension, twist, Mordell-Weil group, Selmer group, Tate-Shafarevitch group, Birch and Swinnerton-Dyer conjecture, local norm index. 'Research partially supported by a grant from the National Science Foundation. ? 1981 American Mathematical Society 0002-9947/81 /0000-0108/$04.75 121122 KENNETH KRAMER we hope to resolve it in the future. (See [8, ?4] for a general discussion of local norm problems.) The standard