Counting polarizations on abelian varieties with group action

TL;DR

This paper investigates how many principal polarizations exist on abelian varieties with group actions, revealing that the count can be greater than one, which challenges previous assumptions of uniqueness.

Contribution

It provides a detailed analysis of the number of principal polarizations on abelian varieties with automorphism group actions, showing that this number can vary and is not always one.

Findings

Number of principal polarizations can be greater than one.

The count varies depending on the abelian variety and its automorphisms.

Challenges the assumption of unique polarization in certain cases.

Abstract

Let $\mathcal{A}_g$ be the moduli space of principally polarized abelian varieties. We study the problem of counting the number of principal polarizations modulo the natural action of the automorphism group of the abelian variety on a very general element of a positive dimensional component of $\mathrm{Sing}(\mathcal{A}_g)$, and show that this number is not always 1.