Strongly compatible systems associated to semistable abelian varieties

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 $\mathrm{G}\mathrm{a}\mathrm{l}(\bar{\mathrm{E}}\mathrm{/}\mathrm{E})$ on the $\mathrm{\ell }$-adic cohomology ${\mathrm{H}}_{\acute{\mathrm{e}}\mathrm{t}}^1(A_{\bar{\mathrm{E}}},{\mathbb{Q}}_{\mathrm{\ell }})$ gives rise to a strongly compatible system of $\mathrm{\ell }$-adic representations valued in the Mumford–Tate group $\mathbf{G}$ of $A$. This involves an independence of $\mathrm{\ell }$-statement for the Weil–Deligne representation associated to $A$ at places of semistable reduction, extending previous work of ours at places of good reduction.