Special values of L-functions of one-motives over function fields

TL;DR

This paper derives a formula for the leading coefficient of L-functions of one-motives over function fields using Weil-étale cohomology, extending the BSD conjecture and relating Tamagawa numbers to cohomology.

Contribution

It generalizes the Weil-étale version of the BSD conjecture for one-motives and expresses Tamagawa numbers in terms of Weil-étale cohomology.

Findings

Formula for the leading coefficient at s=1 of L-functions of one-motives

Expression of Tamagawa numbers via Weil-étale cohomology

Reproof of Ono-Oesterlé Tamagawa number formula

Abstract

The purpose of this paper is to give a formula for the leading coefficient at $s=1$ of the $L$-function of one-motives over function fields in terms of Weil-\'etale cohomology, generalizing the Weil-\'etale version of the Birch and Swinnerton-Dyer conjecture in the authors' previous work. As a consequence we express the Tamagawa number of a torus introduced by Ono-Oesterl\'e in terms of Weil-\'etale cohomology, and reprove their Tamagawa number formula.