The Galois Module Structure of ℓ-adic Realizations of Picard 1-motives and Applications

Abstract

Let Z→Z′ be a finite G-Galois cover of smooth, projective curves over an arbitrary algebraically closed field κ, and let and be G-equivariant, disjoint, finite, non-empty sets of closed points on Z, such that contains the ramification locus of the cover. In this context, we prove that the ℓ-adic realizations of the Picard 1-motive associated to the data are G-cohomologically trivial and therefore -projective modules of finite rank, for all prime numbers ℓ. In the process, we give a new proof of Nakajima’s theorem [Equivariant form of the Deuring–Safarevic formula for Hasse–Witt invariants. Mathematische Zeitschrift 190, no. 4 (1985): 559–66] on the Galois module structure of the semi-simple piece under the action of the Cartier operator on a certain space of differentials associated to Z and ⁠, assuming that char(κ)=p. As a main arithmetic application of these results, we consider the situation where the set of data is defined over a finite field and ⁠. We combine results of Deligne and Tate, Berthelot, Bloch and Illusie with our cohomological triviality result to prove that in this context we have an equality of -ideals ⁠, for all prime numbers ℓ, where ⁠, γ is the q-power arithmetic Frobenius morphism (viewed as a distinguished topological generator of ⁠) and is the (polynomial) G-equivariant L-function associated to the data ⁠. We obtain this way a Galois-equivariant refinement of results of Deligne and Tate [Les conjectures de Stark sur les fonctions L d’Artin en s=0. Progress in Mathematics 47. Boston, MA: Birkhauser Boston Inc., 1984. Lecture notes edited by Dominique Bernardi and Norbert Schappacher.] on ℓ-adic realizations of Picard 1-motives associated to (global) function fields. As an immediate application, we prove refinements of the classical Brumer-Stark and Coates-Sinnott conjectures linking special values of to certain invariants of ideal-class groups and étale cohomology groups, respectively. In our upcoming work, we will show how several other classical conjectures on special values of global L-functions follow from the results obtained in this paper.