Special Values of the Zeta Function of an Arithmetic Surface

TL;DR

This paper investigates the special value conjecture for the Zeta function of arithmetic surfaces, computes correction factors, and links the conjecture to the Birch and Swinnerton-Dyer Conjecture, advancing understanding in arithmetic geometry.

Contribution

It computes the correction factor C(X,1) for the Zeta function and establishes the equivalence of the Flach-Morin conjecture with the Birch and Swinnerton-Dyer Conjecture for arithmetic surfaces.

Findings

Computed the correction factor C(X,1) for the Zeta function.

Established the equivalence between the Flach-Morin conjecture and BSD conjecture for surfaces.

Developed results on the eh-topology relevant to the conjecture.

Abstract

We study the special value conjecture for the Zeta function of a proper regular arithmetic scheme X introduced by Flach and Morin in the case n=1. We compute the correction factor C(X,1) left unspecified in the original statement of the Flach-Morin Conjecture, thereby developing some results on the eh-topology introduced by Geisser. We then specialize further to the case where X is an arithmetic surface and show that the conjecture of Flach and Morin is equivalent to the Birch and Swinnerton-Dyer Conjecture.