Strongly compatible systems associated to abelian varieties
Strongly compatible systems associated to abelian varieties
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Rong Zhou, University of Cambridge
IAS - Simonyi Hall 101
Let $A$ be an abelian variety over a number field $\rmE\subset \bbC$. We prove that after replacing $\rmE$ by a finite extension, the action of $\mathrm{Gal}(\overline{\rmE}/\rmE)$ on the $\ell$-adic Tate modules of $A$ gives rise to a strongly compatible system of $\ell$-adic representations valued in the Mumford--Tate group $\bfG$ 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. This is joint work with Mark Kisin.