Integrality of Seshadri constants and irreducibility of principal polarizations on products of two isogenous elliptic curves

TL;DR

This paper characterizes when all Seshadri constants on a product of two isogenous elliptic curves are integers, linking this to the minimal degree of isogenies and providing a criterion for irreducible principal polarizations.

Contribution

It translates the problem of Seshadri constants into a quadratic form analysis and characterizes when these constants are integers based on isogeny degrees.

Findings

All Seshadri constants are integers if and only if the minimal isogeny degree is 1 or 2.

Provides a numerical criterion for the integrality of Seshadri constants.

Characterizes irreducible principal polarizations on the product of two elliptic curves.

Abstract

In this paper we consider the question of when all Seshadri constants on a product of two isogenous elliptic curves $E_1\times E_2$ without complex multiplication are integers. By studying elliptic curves on $E_1\times E_2$ we translate this question into a purely numerical problem expressed by quadratic forms. By solving that problem, we show that all Seshadri constants on $E_1\times E_2$ are integers if and only if the minimal degree of an isogeny $E_1\to E_2$ equals 1 or 2. Furthermore, this method enables a characterization of irreducible principal polarizations on $E_1\times E_2$.