On the parity of supersingular Weil polynomials

TL;DR

This paper proves that for supersingular abelian varieties over finite fields with certain conditions, the characteristic polynomial of the Frobenius endomorphism is even, extending known results from elliptic curves to higher dimensions.

Contribution

It generalizes the parity property of Frobenius polynomials from supersingular elliptic curves to higher-dimensional supersingular abelian varieties under specific prime conditions.

Findings

Characteristic polynomial is even when p > 2g+1.

Extends parity results from elliptic curves to higher dimensions.

Provides conditions for the polynomial's symmetry in supersingular cases.

Abstract

Let q be an odd power of a prime p and let A/Fq be a supersingular abelian variety of dimension g. We show that if p>2g+1, then the characteristic polynomial of the q-Frobenius is an even polynomial. This generalizes the well-known result on the vanishing of the trace of the p-Frobenius when p>3 for supersingular elliptic curves over Fp.