Angle ranks of abelian varieties

Using the formalism of Newton hyperplane arrangements, we resolve the open questions regarding angle rank left over from work of the first two authors with Roe and Vincent. As a consequence we end up generalizing theorems of Lenstra-Zarhin and Tankeev proving several new cases of the Tate conjecture for abelian varieties over finite fields. We also obtain an effective version of a recent theorem of Zarhin bounding the heights of coefficients in multiplicative relations among Frobenius eigenvalues.