Prosaic Abelian Varieties Bad at One Prime

We say that an abelian variety $A_{\mathrm{/}\mathbf{Q}}$ 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 $p\equiv 18$, 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 ${\mathrm{\Xi }}_p$ 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 $p\equiv 18$ 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 ${\mathrm{\Xi }}_p$. We thereby show that $2g+2\le h_2$ and $p$ has the form $a^2+16b^2$, with $a+4b\equiv \pm 18$. Moreover, if $2g+4\le h_2$, then $p$ has the form $a^2+64b^2$, with $a\equiv \pm 18$.