On the existence of abelian surfaces with everywhere good reduction

Let $D\le 2000$ be a positive discriminant such that $F = $\mathbf{Q}$(\sqrt{D})$hasnarrowclassone,and$A/F$anabeliansurfaceof${\rm GL}_2$-typewitheverywheregoodreduction\mathrm{.}Assumingthat$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 ${\mathrm{E}\mathrm{n}\mathrm{d}}_{\mathbf{Q}}(B)=\mathbf{Z}$, 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.