Joint IAS/PU Arithmetic Geometry

Let A be an abelian variety over a number field E ⊂ ℂ. We prove that, after replacing E by a finite extension, the action of Gal(Ē/E) on the ℓ‑adic Tate modules of A gives rise to a strongly compatible system of ℓ‑adic representations valued in the Mumford–Tate group G of A. This involves an ℓ‑independence statement for the Weil–Deligne representation associated to A at places of semistable reduction, extending our previous work at places of good reduction. This is joint work with Mark Kisin.