Parity conjecture for abelian surfaces

TL;DR

This paper proves that, under certain assumptions, the Birch--Swinnerton-Dyer conjecture accurately predicts the parity of the rank for specific abelian surfaces, advancing understanding in number theory and algebraic geometry.

Contribution

It establishes the parity conjecture for semistable principally polarised abelian surfaces assuming the finiteness of the Tate--Shafarevich group, with additional conditions for Jacobians.

Findings

Parity of rank predicted by BSD matches actual rank parity

Results apply to semistable principally polarised abelian surfaces

Additional conditions required for Jacobians with good ordinary reduction

Abstract

Assuming finiteness of the Tate--Shafarevich group, we prove that the Birch--Swinnerton-Dyer conjecture correctly predicts the parity of the rank of semistable principally polarised abelian surfaces. If the surface in question is the Jacobian of a curve, we require that the curve has good ordinary reduction at 2-adic places.