Angle ranks of abelian varieties

The angle rank of an abelian variety over a finite field (or a CM abelian variety over C) quantifies the extent to which the Tate conjecture (or the Hodge conjecture) holds "for trivial reasons"; cases where this does not happen tend to be rare in practice. Picking up a thread from some old (1980s and 1990s) results of Tankeev and Lenstra-Zarhin, we show that in many cases, the Tate conjecture is forced to hold by the Newton polygon of the abelian variety or the Galois group of the Frobenius eigenvalues. Joint work with Taylor Dupuy and David Zureick-Brown.