Strongly compatible systems associated to semistable abelian varieties

Abstract

We prove a motivic refinement of a result of Weil, Deligne and Raynaud on the existence of strongly compatible systems associated to abelian varieties. More precisely, given an abelian variety $A$ over a number field $\mathrm{E}\subset \mathbb{C}$, we prove that after replacing $\mathbb{E}$ by a finite extension, the action of $\mathrm{Gal}(\overline{\mathrm{E}}/\mathrm{E})$ on the $\ell$-adic cohomology ${\mathrm{H}}_{\acute{\mathrm{e}}\mathrm{t}}^1(A_{\overline{\mathrm{E}}},{\mathbb{Q}}_{\ell })$ gives rise to a strongly compatible system of $\ell$-adic representations valued in the Mumford--Tate group $\mathbf{G}$ of $A$. This involves an independence of $\ell$-statement for the Weil--Deligne representation associated to $A$ at places of semistable reduction, extending previous work of ours at places of good reduction.

44 pages. Comments welcome