On isogenous principally polarized abelian surfaces

TL;DR

This paper explores the relationships between genus 2 curves with isogenous Jacobians, extending classical results for p=2 to the case p=3, and provides explicit descriptions of their Weierstrass points.

Contribution

It generalizes classical results on genus 2 curves and their Jacobians from p=2 to p=3, offering explicit relationships between Weierstrass points.

Findings

Explicit relationship between Weierstrass points for p=3

Extension of Richelot and Humbert classical results

New insights into isogenies of genus 2 Jacobians

Abstract

We study a relationship between two genus 2 curves whose jacobians are isogenous with kernel equal to a maximal isotropic subspace of p-torsion points with respect to the Weil pairing. For p = 3 we find an explicit relationship between the set of Weierstrass points of the two curves extending the classical results of F. Richelot (1837) and G. Humbert (1901) in the case p = 2.