Values of zeta functions of arithmetic surfaces at $s=1$

TL;DR

This paper demonstrates that a recent conjecture regarding the special value of the zeta function at s=1 for arithmetic surfaces is equivalent to the well-known Birch-Swinnerton-Dyer conjecture for the Jacobian of the generic fiber, linking two major conjectures.

Contribution

The paper establishes an equivalence between a new conjecture on zeta functions of arithmetic surfaces and the Birch-Swinnerton-Dyer conjecture, providing a novel perspective on these fundamental problems.

Findings

Equivalence between the zeta function conjecture and BSD conjecture.

Insight into the relationship between arithmetic surfaces and elliptic curves.

Potential implications for proving BSD via zeta function properties.

Abstract

We show that the recent conjecture of the first-named author for the special value at $s=1$ of the zeta function of an arithmetic surface is equivalent to the Birch-Swinnerton-Dyer conjecture for the Jacobian of the generic fibre.