On the existence of abelian surfaces with everywhere good reduction

Abstract

Let $D\le 2000$ be a positive discriminant such that $F=\mathbf{Q}(\sqrt{D})$ has narrow class one, and $A/F$ an abelian surface of ${\mathrm{GL}}_2$-type with everywhere good reduction. Assuming that $A$ is modular, we show that $A$ is either an $F$-surface or is a base change from $\mathbf{Q}$ of an abelian surface $B$ such that ${\mathrm{End}}_{\mathbf{Q}}(B)=\mathbf{Z}$, except for $D=353,421,1321,1597$ and $1997$. In the latter case, we show that there are indeed abelian surfaces with everywhere good reduction over $F$ for $D=353,421$ and $1597$, which are non-isogenous to their Galois conjugates. These are the first known such examples.

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