ON SELMER RANK PARITY OF TWISTS

In this paper we study the variation of the p-Selmer rank parities of p-twists of a principally polarized Abelian variety over an arbitrary number field K and show, under certain assumptions, that this parity is periodic with an explicit period. Our result applies in particular to principally polarized Abelian varieties with full K-rational p-torsion subgroup, arbitrary elliptic curves, and Jacobians of hyperelliptic curves. Assuming the Shafarevich-Tate conjecture, our result allows one to classify the rank parities of all quadratic twists of an elliptic or hyperelliptic curve after a finite calculation.