## Independence of ℓ for Frobenius conjugacy classes attached to abelian varieties

Rong Zhou

Imperial College London

June 18, 2020

Let A be an abelian variety over a number field E⊂ℂ and let v be a place of good reduction lying over a prime p. For a prime ℓ≠p, a result of Deligne implies that upon replacing E by a finite extension, the Galois representation on the ℓ-adic Tate module of A factors as ρℓ:Gal(E⎯⎯⎯⎯/E)→GA, where GA is the Mumford--Tate group of Aℂ. For p>2, we prove that the conjugacy class of ρℓ(Frobv) is defined over ℚ and independent of ℓ. This is joint work with Mark Kisin.