On the Birch-Swinnerton-Dyer quotients modulo squares

Abstract

Let $A$ be an abelian variety over a number field $K$. An identity between the $L$-functions $L(A/K_i,s)$ for extensions $K_i$ of $K$ induces a conjectural relation between the Birch-Swinnerton-Dyer quotients. We prove these relations modulo finiteness of Ш, and give an analogous statement for Selmer groups. Based on this, we develop a method for determining the parity of various combinations of ranks of $A$ over extensions of $K$. As one of the applications, we establish the parity conjecture for elliptic curves assuming finiteness of $Ш(E/K(E[2]))[6^\infty]$ and some restrictions on the reduction at primes above 2 and 3: the parity of the Mordell-Weil rank of $E/K$ agrees with the parity of the analytic rank, as determined by the root number. We also prove the $p$-parity conjecture for all elliptic curves over $\mathbb{Q}$ and all primes $p$: the parities of the $p^\infty$-Selmer rank and the analytic rank agree.