Conference abstracts

Session A3 - Computational Number Theory - Semi-plenary talk

July 12, 15:30 ~ 16:20 - Room B6

Parity of ranks of abelian surfaces

University of Bristol, United Kingdom   -   c.maistret@warwick.ac.uk

Let $K$ be a number field and $A/K$ an abelian surface. By the Mordell-Weil theorem, the group of $K$-rational points on $A$ is finitely generated and as for elliptic curves, its rank is predicted by the Birch and Swinnerton-Dyer conjecture. A basic consequence of this conjecture is the parity conjecture: the sign of the functional equation of the L-series determines the parity of the rank of $A/K$. Under suitable local constraints and finiteness of the Shafarevich-Tate group, we prove the parity conjecture for principally polarized abelian surfaces. We also prove analogous unconditional results for Selmer groups.

FoCM 2017, based on a nodethirtythree design.