On the Birch-Swinnerton-Dyer quotients modulo squares

Details On the Birch-Swinnerton-Dyer quotients modulo squares On the Birch-Swinnerton-Dyer quotients modulo squares

2006-10-09

arxiv.org/abs/math/0610290v3

Mathematics

Number Theory

29 pages; minor changes; to appear in Annals of Mathematics

Scientific paper

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 Sha, 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 Sha[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 Q and all primes p: the parities of the p^\infty-Selmer rank and the analytic rank agree.

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