The Mumford-Tate conjecture asserts that, via the Betti-étale comparison isomorphism, and for any smooth projective variety $ X $, over a number field $ K $, the $ \mathbb{Q}_{ \ell } $-linear combinations of Hodge cycles coincide with the $ \ell $-adic Tate cycles.
Question. Would that mean that if the Hodge conjecture and the Tate conjecture hold, then the Mumford-Tate conjecture holds as well ?
Best Answer
Yes.
Under the Hodge conjecture, the Hodge cycles are the algebraic cycles, so the $\mathbb Q_\ell$-linear combinations of Hodge cycles are the $\mathbb Q_\ell$-linear combinations of algebraic cycles.
Under the Tate conjecture, the $\ell$-adic Tate classes are the $\mathbb Q_\ell$-linear combinations of algebraic cycles.
So under the Hodge and Tate conjectures, these are both equal.
This then implies that the identity component of the $\ell$-adic monodromy group is isomorphic over $\mathbb Q_\ell$ to the Mumford-Tate group, by a Tannakian argument.