Hyperelliptic ${\mathcal{S}}_7$-curves of prime conductor

Abstract

An abelian threefold $A_{/\mathbb{Q}}$ of prime conductor $N$ is favorable if its 2-division field $F$ is an ${\mathcal{S}}_7$-extension over $\mathbb{Q}$ with ramification index 7 over ${\mathbb{Q}}_2$. Let $A$ be favorable and let $B$ be a semistable abelian variety of dimension $3d$ and conductor $N^d$ with $B[2]$ filtered by copies of $A[2]$. We give a sufficient and computable class field theoretic criterion on $F$ to guarantee that $B$ is isogenous to $A^d$.

We thank the anonymous referee for suggestions that were used to improve the organization and exposition in version 2