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

Let Z→Z′ be a finite G-Galois cover of smooth, projective curves over an arbitrary algebraically closed field κ, and let $\mathcal S$ and $\mathcal T$ be G-equivariant, disjoint, finite, non-empty sets of closed points on Z, such that $\mathcal S$ contains the ramification locus of the cover. In this context, we prove that the ℓ-adic realizations $T_\ell (\mathcal M_{\mathcal S,\mathcal T})$ of the Picard 1-motive $M_{\mathcal S,\mathcal T}$ associated to the data $(Z,\kappa,\mathcal S,\mathcal T)$ are G-cohomologically trivial and therefore $\mathbb Z_\ell [G]$-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 $\Omega_Z(-[\mathcal S])^{s}$ under the action of the Cartier operator on a certain space of differentials $\Omega_Z(-[\mathcal S])$ associated to Z and $\mathcal S$, assuming that char(κ)=p. As a main arithmetic application of these results, we consider the situation where the set of data $(Z,\ Z',\ \mathcal S,\ \mathcal T)$ is defined over a finite field $\mathbb F_{q}$ and $\kappa:+\bar {\mathbb F}_{q}$. 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 $\mathbb Z_\ell\left[\left[\bar G\right]\right]$-ideals $(\Theta_\mathcal {S,T}(\gamma^{-1}))={\mathrm Fit}_{\mathbb Z_\ell\left[\left[\bar G\right]\right]}(T_\ell(M_{\mathcal {S,T}}))$, for all prime numbers ℓ, where $\bar G:=G\ \times\ {\mathrm Gal}(\bar {\mathbb F}_q/\mathbb F_q),\ \gamma$ is the q-power arithmetic Frobenius morphism (viewed as a distinguished topological generator of ${\mathrm Gal}(\bar {\mathbb F}_q/\mathbb F_q)$) and $\Theta _{\mathcal {S,T}}(u)\in\mathbb Z[G][u]$ is the (polynomial) G-equivariant L-function associated to the data $(Z,\ Z',\ \mathbb F_{q},\ \mathcal {S,\ T})$. 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 $\Theta_\mathcal {S,T}(u)$ 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.