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Let  Z → Z ′ be a finite  G -Galois cover of smooth, projective curves over an arbitrary algebraically closed field  κ , and letandbe  G -equivariant, disjoint, finite, non-empty sets of closed points on  Z , such thatcontains the ramification locus of the cover. In this context, we prove that the ℓ-adic realizationsof the Picard 1-motiveassociated to the dataare  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 pieceunder the action of the Cartier operator on a certain space of differentialsassociated to  Z  and, assuming that char( κ )= p . As a main arithmetic application of these results, we consider the situation where the set of datais defined over a finite fieldand. 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) andis 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 ofto 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.

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