3 problems

$\ell$-Mumford–Tate conjecture. This map identifies the $\mathbb{Q}_{\ell}$-vector spaces spanned by Hodge classes and Tate classes. The conjecture expresses the expected equivalen…

Integral Mumford–Tate conjecture. There is a constant $C$ such that, for all primes $\ell$, the subgroup $\rho_\ell(\Gamma_{k^{\rm conn}})$ has index at most $C$ in…

Let $A$ be a non-zero abelian variety over a number field $K$ satisfying $K_A^{\operatorname{conn}}=K$. Let $\mathcal W_{A_v}$ be the set of roots of the Frobenius characteristic p…