The refined class number formula for Drinfeld modules

TL;DR

This paper refines Taelman's class number formula for Drinfeld modules over Galois extensions of function fields, providing explicit Galois module structure results for the associated class groups.

Contribution

It presents an equivariant refinement of Taelman's analytic class number formula for Drinfeld modules, extending understanding of Galois structures in this context.

Findings

Refined the class number formula for Drinfeld modules

Derived explicit Galois module structure of Taelman class groups

Extended Taelman's results to equivariant settings

Abstract

Let $K/k$ be a finite Galois extension of global function fields. Let $E$ be a Drinfeld module over $k$. We state and prove an equivariant refinement of Taelman's analogue of the analytic class number formula for $(E,K/k)$, and derive explicit consequences for the Galois structure of the Taelman class group of $E$ over $K$.