Isogeny classes of non-simple abelian surfaces over finite fields

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.