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 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.

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

References in corpus (4)

• Regulator constants and the parity conjecture
• Computations in non-commutative Iwasawa theory
• Parity of ranks for elliptic curves with a cyclic isogeny
• Self-duality of Selmer groups

Cited by in corpus (16)

• Root numbers and parity of ranks of elliptic curves
• The average size of the 3-isogeny Selmer groups of elliptic curves $y^2=x^3+k$
• Density of rational points on isotrivial rational elliptic surfaces
• Growth of Sha in towers for isogenous curves
• Rank growth of elliptic curves over $N$-th root extensions
• Root numbers and parity of local Iwasawa invariants
• The 2-parity conjecture for elliptic curves with isomorphic 2-torsion
• Comparing local constants of elliptic curves in dihedral extensions
• On Brauer-Kuroda type relations of S-class numbers in dihedral extensions
• Quadratic Twists of Elliptic Curves with 3-Selmer Rank 1
• Parity conjectures for elliptic curves over global fields of positive characteristic
• Duality invariance of Faltings heights, Hodge line bundles and global periods
• Rank growth of elliptic curves in $S_4$ and $A_4$ quartic extensions of the rationals
• Factor equivalence of Galois modules and regulator constants
• Relations between permutation representations in positive characteristic
• On central $L$-values and the growth of the $3$-part of the Tate-Shafarevich group