On a Birch and Swinnerton-Dyer type conjecture for the Hasse-Weil-Artin $L$-functions in characteristic $p>0$

TL;DR

This paper establishes a Birch and Swinnerton-Dyer type formula for Hasse-Weil-Artin L-functions in characteristic p>0, extending prior work to include cases with wild ramification and using equivariant Riemann-Roch theory.

Contribution

It provides a new BSD-type formula for L-functions associated with abelian varieties and Galois representations over function fields, including the p-part under wild ramification.

Findings

Derived a BSD-type formula for L-functions in characteristic p>0.

Extended applicability to cases with wild ramification.

Utilized equivariant Riemann-Roch theorem for vector bundles.

Abstract

Given an abelian variety $A$ over a global function field $K$ of characteristic $p>0$ and an irreducible complex continuous representation $\psi$ of the absolute Galois group of $K$, we obtain a BSD-type formula for the leading term of Hasse--Weil--Artin $L$-function for $(A,\psi)$ at $s=1$ under certain technical hypotheses. The formula we obtain can be applied quite generally; for example, it can be applied to the $p$-part of the leading term even when $\psi$ is weakly wildly ramified at some place under additional hypotheses. Our result is the function field analogue of the work of D. Burns and D. Macias Castillo, built upon the work on the equivariant refinement of the BSD conjecture by D. Burns, M. Kakde and the first-named author. To handle the $p$-part of the leading term, we need the Riemann--Roch theorem for equivariant vector bundles on a curve over a finite field generalising the work of S. Nakajima, B. K\"ock, and H. Fischbacher-Weitz and B. K\"ock, which is of independent interest.