Small degree isogenies between conjugate principally polarized superspecial abelian surfaces

Abstract

Let $p>3$ be a prime number. We prove that every principally polarized superspecial abelian surface over ${\mathbb{F}}_{p^2}$ with $p^2$-Frobenius $[-p]$ admits a separable polarized isogeny to its Frobenius conjugate with multiplier at most $(p^3/2{)}^{1/5}$. This generalizes a recent result of Aubry, Oyono, and Vincent (2026, arXiV:2607.14624) who proved that every supersingular elliptic curve defined over ${\bar{\mathbb{F}}}_p$ admits an isogeny to its Frobenius conjugate with degree at most $(p/2{)}^{1/3}$.