Isogeny classes of non-simple abelian surfaces over finite fields

TL;DR

This paper investigates the number of principally polarized abelian surfaces over finite fields that are isogenous to a given split abelian surface, providing asymptotic bounds that refine previous results.

Contribution

It offers refined asymptotic bounds on the count of isogenous principally polarized abelian surfaces over finite fields, extending prior work by Achter and Howe.

Findings

Established asymptotic upper bounds

Established asymptotic lower bounds

Refined previous estimates on isogeny classes

Abstract

Let $A=E \times E_{ss}$ be a principally polarized almost ordinary split abelian surface over a finite field $\mathbb{F}_{q}$. We give asymptotic upper and lower bounds on the number of principally polarized abelian surfaces over $\mathbb{F}_{q^{n}}$ that are $\overline{\mathbb{F}}_{q}$-isogenous to $A$ up to isomorphism, which is a refinement of the results in the work of Achter and Howe.