# Tamagawa number formula with coefficients over varieties in positive characteristic

## TL;DR

This paper generalizes the Tamagawa number formula to include coefficients over varieties in positive characteristic, utilizing F-gauges and Raynaud modules to analyze L-functions.

## Contribution

It extends the Tamagawa number formula to a broader class of coefficients over varieties in characteristic p, introducing new methods involving F-gauges and Raynaud modules.

## Key findings

- Expressed the order of the pole of L-functions in positive characteristic.
- Determined the leading coefficient of L-functions for a large class of coefficients.
- Generalized previous results by Milne-Ramachandran with new techniques.

## Abstract

We express the order of the pole and the leading coefficient of the L-function of a (large class of) -adic coefficients (any prime) over a quasi-projective variety over a finite field of characteristic p. This is a generalization of the result of Milne-Ramachandran with coefficients. The new key ingredient is the use of F-gauges and their equivalence in the derived category with Raynaud modules proved by Ekedahl.