Certain Abelian varieties bad at only one prime

TL;DR

This paper investigates abelian surfaces with prime conductor and specific ramification properties, providing criteria to determine when related abelian varieties are isogenous, and identifying unique isogeny classes for certain conductors.

Contribution

It introduces a class field theoretic criterion for isogeny of abelian varieties based on their 2-division fields and ramification, supporting the paramodular conjecture.

Findings

Unique isogeny class for each conductor in {277, 349, 461, 797, 971}

Criterion applies to abelian varieties with specific ramification properties

Provides data on the general applicability of the criterion

Abstract

An abelian surface $A_{/{\mathbb Q}}$ of prime conductor $N$ is favorable if its 2-division field $F$ is an ${\mathcal S}_5$-extension with ramification index 5 over ${\mathbb Q}_2$. Let $A$ be favorable and let $B$ be any semistable abelian variety of dimension $2d$ and conductor $N^d$ such that $B[2]$ is filtered by copies of $A[2]$. We give a sufficient class field theoretic criterion on $F$ to guarantee that $B$ is isogenous to $A^d$. As expected from our paramodular conjecture, we conclude that there is one isogeny class of abelian surfaces for each conductor in $\{277, 349,461,797,971\}$. The general applicability of our criterion is discussed in the data section.