The refined class number formula for Drinfeld modules

Let $\mathbb{F}_q$ be a finite field, let $K/k$ be a finite Galois extension of function fields over $\mathbb{F}_q$, and let $E$ be a Drinfeld $\mathbb{F}_q[t]$-module defined over the ring of integers of $k$. In joint work with Daniel Macías Castillo and Daniel Martínez Marqués, we have formulated and proven an equivariant refinement of Taelman's formula for the special value of the Goss L-function attached to a Drinfeld module (which can be interpreted as a function field analogue of the analytic class number formula).

In this talk, I will review Taelman's work and discuss our formulation of the equivariant class number formula, as well as some explicit consequences for the Galois module structure of the Taelman class group of $E$ over $K$.