An equivariant Tamagawa number formula for Drinfeld modules and beyond

Abstract: To a Galois extension of characteristic p global fields and a suitable Drinfeld module, one can associate an equivariant, characteristic p valued, rigid analytic Goss-type L-function. I will discuss the construction of this L-function and the statement and proof of a (Tamagawa number) formula for its special value at 0, which generalizes to the Galois equivariant setting Taelman's celebrated class-number formula, proved in 2012. Next, I will show how this formula implies a perfect analog of the Brumer-Stark conjecture for Drinfeld modules. If time permits, I will discuss the very recent extension of the above formula to the much larger category of t-modules (t-motives), as well as its applications to the development of an Iwasawa theory for Drinfeld modules. The lecture is based on several recent joint works with N. Green, J. Ferrara and Z. Higgins.

algebraic geometrynumber theory

Audience: researchers in the topic

UCSB Seminar on Geometry and Arithmetic