Principally polarizable isogeny classes of abelian surfaces over finite fields

TL;DR

This paper characterizes when an isogeny class of abelian surfaces over finite fields contains a principally polarizable surface, linking algebraic conditions on the Weil polynomial to the existence of principal polarizations.

Contribution

It provides a precise criterion based on the Weil polynomial parameters for the existence of principal polarizations in isogeny classes of abelian surfaces.

Findings

A class contains a principally polarizable surface iff a^2 - b ≠ q or b ≥ 0 or some prime divisor of b ≠ 1 mod 3.

The paper establishes a necessary and sufficient condition involving the parameters a and b of the Weil polynomial.

The results are used to determine which classes contain Jacobians in a subsequent work.

Abstract

Let A be an isogeny class of abelian surfaces over F_q with Weil polynomial x^4 + ax^3 + bx^2 + aqx + q^2. We show that A does not contain a surface that has a principal polarization if and only if a^2 - b = q and b < 0 and all prime divisors of b are congruent to 1 modulo 3. We use this result in a forthcoming paper in which we determine which isogeny classes of abelian surfaces over finite fields contain Jacobians.