# On the existence of abelian surfaces with everywhere good reduction

## TL;DR

This paper investigates the existence and classification of abelian surfaces with everywhere good reduction over quadratic fields with certain discriminants, providing new explicit examples and extending understanding of their modularity and endomorphism properties.

## Contribution

It proves that under certain conditions, such abelian surfaces are either base changes from Q or are defined over the field, except for specific discriminants, and constructs explicit examples for some cases.

## Key findings

- Classifies abelian surfaces with good reduction over quadratic fields for certain discriminants.
- Shows existence of non-isogenous abelian surfaces with everywhere good reduction for specific discriminants.
- Provides the first known explicit examples of such surfaces over these fields.

## 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.