Special L-values of abelian t-modules

TL;DR

This paper establishes a formula for the special L-values of abelian t-modules over function fields, providing analogues of the class number formula and linking to shtuka extension groups.

Contribution

It introduces a new formula for $ abla$-adic special L-values of abelian t-modules, connecting them with extension groups of shtukas, advancing the understanding of function field arithmetic.

Findings

Derived a formula for $ abla$-adic special L-values of abelian t-modules

Established analogues of the class number formula in the function field setting

Connected special L-values with extension groups of shtukas

Abstract

We prove a formula for the $\infty$-adic special $L$-value of abelian $t$-modules. This gives function field analogues of the class number formula. We also express it in terms of the extension groups of shtukas.