Vol. 12, No. 5, 2018

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Certain abelian varieties bad at only one prime

Armand Brumer and Kenneth Kramer

Vol. 12 (2018), No. 5, 1027–1071
Abstract

An abelian surface ${A}_{∕ℚ}$ of prime conductor $N$ is favorable if its 2-division field $F$ is an ${\mathsc{S}}_{5}$-extension over $ℚ$ with ramification index 5 over ${ℚ}_{2}$. Let $A$ be favorable and let $B$ be a semistable abelian variety of dimension $2d$ and conductor ${N}^{d}$ with $B\left[2\right]$ filtered by copies of $A\left[2\right]$. 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 $\left\{277,349,461,797,971\right\}$. The general applicability of our criterion is discussed in the data section.

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