Euler factors of equivariant $L$--functions of Drinfeld modules and beyond

TL;DR

This paper proves identities involving Euler factors of equivariant L-functions for Drinfeld modules, extending previous results from the Carlitz module to general Drinfeld modules and suggesting applications to higher-dimensional abelian t-modules.

Contribution

It develops techniques to verify Euler factor identities for arbitrary Drinfeld modules, generalizing prior work on the Carlitz module and indicating extensions to abelian t-modules.

Findings

Proved Euler factor identities for general Drinfeld modules.

Extended techniques to higher-dimensional abelian t-modules.

Supported the equivariant Tamagawa number formula in broader contexts.

Abstract

In \cite{FGHP}, the first author and his collaborators proved an equivariant Tamagawa number formula for the special value at $s=0$ of a Goss--type $L$--function, equivariant with respect to a Galois group $G$, and associated to a Drinfeld module defined on $\Bbb F_q[t]$ and over a finite, integral extension of $\Bbb F_q[t]$. The formula in question was proved provided that the values at $0$ of the Euler factors of the equivariant $L$--function in question satisfy certain identities involving Fitting ideals of certain $G$--cohomologically trivial, finite $\Bbb F_q[t][G]$--modules associated to the Drinfeld module. In \cite{FGHP}, we prove these identities in the particular case of the Carlitz module. In this paper, we develop general techniques and prove the identities in question for arbitrary Drinfeld modules. Further, we indicate how these techniques can be extended to the more general case of higher dimensional abelian $t$--modules, which is relevant in the context of the proof of the equivariant Tamagawa number formula for abelian $t$--modules given by N. Green and the first author in \cite{Green-Popescu}. This paper is based on a lecture given by the first author at ICMAT Madrid in May 2023 and builds upon results obtained by the second author in his PhD thesis \cite{Ramachandran-thesis}.