Angle ranks of abelian varieties

TL;DR

This paper uses Newton hyperplane arrangements to resolve open questions about angle ranks of abelian varieties, leading to new cases of the Tate conjecture and bounds on Frobenius eigenvalues.

Contribution

It generalizes previous theorems and proves new cases of the Tate conjecture for abelian varieties over finite fields.

Findings

Resolved open questions on angle ranks using Newton hyperplane arrangements

Proved several new cases of the Tate conjecture for abelian varieties over finite fields

Provided an effective bound on heights of coefficients in relations among Frobenius eigenvalues

Abstract

Using the formalism of Newton hyperplane arrangements, we resolve the open questions regarding angle rank left over from [DKRV20]. 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.