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

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.