Arithmetic of elliptic curves upon quadratic extension

Abstract

This paper is a study of variations in the rank of the Mordell-Weil group of an elliptic curve E E defined over a number field F F as one passes to quadratic extensions K K of F F . Let S ( K ) S(K) be the Selmer group for multiplication by 2 2 on E ( K ) E(K) . In analogy with genus theory, we describe S ( K ) S(K) in terms of various objects defined over F F and the local norm indices i υ = dim F 2 E ( F υ ) / Norm { E ( K w ) } {i_\upsilon } = {\dim _{{{\mathbf {F}}_2}}}E({F_\upsilon })/\text {Norm} \{ E({K_w})\} for each completion F υ {F_\upsilon } of F F . In particular we show that dim ⁡ S ( K ) + dim ⁡ E ( K ) 2 \dim S(K) + \dim E{(K)_2} has the same parity as Σ i υ \Sigma {i_\upsilon } . We compute i υ {i_\upsilon } when E E has good or multiplicative reduction modulo υ \upsilon . Assuming that the 2 2 -primary component of the Tate-Shafarevitch group Ш ( K ) \Sha (K) is finite, as conjectured, we obtain the parity of rank E ( K ) E(K) . For semistable elliptic curves defined over Q {\mathbf {Q}} and parametrized by modular functions our parity results agree with those predicted analytically by the conjectures of Birch and Swinnerton-Dyer.