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 $E$ 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 ${i_\upsilon } = {\dim _{{{\mathbf {F}}_2}}}E({F_\upsilon })/\text {Norm} \{ E({K_w})\}$ 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 $\Sha (K)$ 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.