Parity conjecture for abelian surfaces

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.

arXiv commentAppendix A by A. Morgan and Appendix B by T. Dokchitser and V. Dokchitser. Minor changes