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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Arithmetic of elliptic curves upon quadratic extension

Author: Kenneth Kramer
Journal: Trans. Amer. Math. Soc. 264 (1981), 121-135
MSC: Primary 14G25; Secondary 10B10, 14K07
MathSciNet review: 597871
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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 $ {i_\upsilon } = {\dim _{{{\mathbf{F}}_2}}}E({F_\upsilon })/$Norm$ \{ E({K_w})\}$ for each completion $ {F_\upsilon }$ of $ F$. In particular we show that $ \dim S(K) + \dim E{(K)_2}$ has the same parity as $ \Sigma {i_\upsilon }$. We compute $ {i_\upsilon }$ when $ E$ has good or multiplicative reduction modulo $ \upsilon $. Assuming that the $ 2$-primary component of the Tate-Shafarevitch group $ \mathcyr{SH}(K)$ is finite, as conjectured, we obtain the parity of rank $ E(K)$. For semistable elliptic curves defined over $ {\mathbf{Q}}$ and parametrized by modular functions our parity results agree with those predicted analytically by the conjectures of Birch and Swinnerton-Dyer.

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Keywords: Elliptic curve, quadratic extension, twist, Mordell-Weil group, Selmer group, Tate-Shafarevitch group, Birch and Swinnerton-Dyer conjecture, local norm index
Article copyright: © Copyright 1981 American Mathematical Society