Fitting ideals of ℓ-adic realizations of Picard 1-motives and class groups of global function fields

Abstract

In previous work, we proved a result on the equivariant Fitting ideal of the `-adic realization of the Picard 1-motive attached to an abelian covering of curves defined over a finite field. In this paper, we build upon this work to deduce results on the equivariant Fitting ideal of the Tate modules of the Jacobian of the top curve, and on the equivariant Fitting ideal of the (dual of the) degree zero class group of the top curve. At the end, we discuss refinements of Brumer’s conjecture and consider examples.