On Frobenius traces

In this paper we discuss a certain Diophantine property of Frobenius traces associated with an Abelian variety over a number field $k$ and apply it to prove the Mumford–Tate conjecture for 4$p$-dimensional Abelian varieties $J$ over $k$, where $p$ is a prime number, $p\geqslant 17$, or (under certain weak assumptions) $\operatorname{End}^0(J\otimes\overline k)$ is an imaginary quadratic extension of $\mathbb Q$.