The conjectures of Artin-Tate and Birch-Swinnerton-Dyer

TL;DR

This paper proves the equivalence of the Artin-Tate and Birch-Swinnerton-Dyer conjectures for fibered surfaces and their Jacobians, offering new insights and proofs in algebraic geometry and number theory.

Contribution

It establishes the equivalence of two major conjectures and provides a new proof of a related theorem, advancing understanding in algebraic geometry.

Findings

Proved the equivalence of Artin-Tate and BSD conjectures for fibered surfaces.

Provided a new proof of Geisser's theorem relating Brauer and Tate-Shafarevich groups.

Enhanced theoretical understanding of conjectures in algebraic geometry.

Abstract

We provide two proofs that the conjecture of Artin-Tate for a fibered surface is equivalent to the conjecture of Birch-Swinnerton-Dyer for the Jacobian of the generic fibre. As a byproduct, we obtain a new proof of a theorem of Geisser relating the orders of the Brauer group and the Tate-Shafarevich group.