Prosaic Abelian Varieties Bad at One Prime

Abstract

We say that an abelian variety $A_{/\mathbf Q}$ of dimension $g$ is {\em prosaic} if it is semistable, with good reduction at 2 and its points of order $2$ generate a $2$-extension of ${\mathbf Q}$. For $p \equiv 1 \bmod{8}$, let $M_u$ be the maximal 2-primary unramified abelian extension of $K = {\mathbf Q}(\sqrt{-p})$ and let $h_2 =[M_u:K]$. We construct an indecomposable group scheme $\Xi_p$ over ${\mathbf Z}[\frac{1}{p}]$ of exponent 2 with field of points $M_u$. Assume that $A$ is prosaic, with bad reduction at only one prime $p$. Then $p \equiv 1 \bmod{8}$ and $A$ is totally toroidal at $p$. We prove that if End $A={\mathbf Z}$, then there is a ${\mathbf Q}$-isogenous abelian variety $B$ such that $B[2]$ is a subquotient of $\Xi_p$. We thereby show that $2g+2 \le h_2$ and $p$ has the form $a^2+16b^2$, with $a+4b \equiv \pm 1 \bmod{8}$. Moreover, if $2g + 4 \le h_2$, then $p$ has the form $a^2+64b^2$, with $a \equiv \pm 1 \bmod{8}$.