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

Abstract

In [Algebra Number Theory 16 (2022), 2215–2264], 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 ${\mathbb{F}}_q[t]$ and over a finite, integral extension of ${\mathbb{F}}_q[t]$. 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 ${\mathbb{F}}_q[t][G]$-modules associated to the Drinfeld module. In op. cit., 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 Chapter 10 of the current volume.