Certain Abelian varieties bad at only one prime

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.

In version 3, the exposition is further improved

References in corpus (1)

• Theta Lifts of Bianchi Modular Forms and Applications to Paramodularity

Cited by in corpus (1)

• On the paramodularity of typical abelian surfaces (and reduction of G-covariant bilinear forms)