The Galois module structure of l-adic realizations of Picard 1-motives and applications

TL;DR

This paper investigates the Galois module structure of l-adic realizations of Picard 1-motives associated with Galois covers of curves, generalizing Nakajima's theorem and applying to conjectures in number theory.

Contribution

It generalizes Nakajima's theorem to a broader setting, computes Fitting ideals in abelian cases, and refines conjectures like Brumer-Stark for global function fields.

Findings

G-cohomological triviality of l-adic realizations for all primes l

Explicit computation of Fitting ideals in abelian cases

Refined versions of Brumer-Stark and Coates-Sinnott conjectures

Abstract

We show that the l-adic realizations of certain Picard 1-motives associated to a G-Galois cover of smooth, projective curves defined over an algebraically closed field are G-cohomologically trivial, for all primes l. In the process, we generalize a well-known theorem of Nakajima on the Galois module structure of certain spaces of Kahler differentials associated to the top curve of the cover, assuming that the field of definition is of characteristic p. If the cover and the Picard 1-motive are defined over a finite field and if G is abelian, we compute the first Fitting ideal of each l-adic realization over a certain profinite l-adic group algebra in terms of an equivariant Artin L-function. This is a refinement of earlier work of Deligne and Tate on Picard 1-motives associated to global function fields. As a consequence, we prove refined versions of the Brumer-Stark and the (\'etale) Coates-Sinnott Conjectures for global function fields. In upcoming work, we use these results to prove several other conjectures on special values of global L-functions in characteristic p. Also, we prove analogous results in characteristic 0, where the l-adic realizations of Picard 1-motives will be replaced by a new class of Iwasawa modules.