##### Department of Mathematics,

University of California San Diego

****************************

### Math 209 - Number Theory

## Joseph Ferrara and Cristian Popescu

#### UCSD

## An equivariant Tamagawa number formula for Drinfeld modules I, II

##### Abstract:

In this series of lectures we will describe the construction of a $G$-equivariant $L$-function $Theta^E_{K/F}(s)$ associated to an abelian extension $K/F$ of characteristic $p$ global fields of Galois group $G$ and a suitable Drinfeld module $E$ defined over $F$, as well as state and sketch the proof of a theorem linking the special value $Theta^E_{K/F}(0)$ to a quotient of volumes of certain compact topological spaces canonically associated to the pair $(K/F, E)$. In lecture I (1-2pm), Cristian will define the $L$--function, give an arithmetic interpretation of its special value at $s=0$ and state the main theorem. In lecture II (2-3pm), Joe will introduce the main ingredients involved in the proof of the main theorem and sketch the main ideas of proof. These lectures describe joint work of J. Ferrara, N. Green, Z. Higgins and C. Popescu. The results within generalize to the Galois equivariant setting earlier work of L. Taelman on special values of Goss zeta functions associated to Drinfeld modules (Taelman, Annals of Math. 2010).

### November 7, 2019

### 12:00 PM

### AP&M 7321

****************************