Strongly compatible systems associated to semistable abelian varieties

TL;DR

This paper proves that for a semistable abelian variety over a number field, the Galois action on its étale cohomology forms a strongly compatible system of representations, refining classical results with motivic insights.

Contribution

It establishes the existence of strongly compatible systems associated to abelian varieties at places of semistable reduction, extending prior results to a broader class of reduction types.

Findings

Existence of strongly compatible systems after finite extension of the base field.

Independence of ℓ for Weil-Deligne representations at semistable places.

Extension of previous good reduction results to semistable reduction cases.

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 $\mathrm H^1_{\mathrm{\acute{e}t}}(A_{\overline{\mathrm E}},\mathbb Q_\ell)$ 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.