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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
${\rm GL}_2$-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 ${\rm End}_{\mathbf{Q}}(B) = \mathbf{Z}$,
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.