## On the existence of abelian surfaces with everywhere good reduction

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Lassina Dembélé
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## Abstract:

Let $D \le 2000$ be a positive discriminant such that $F = \mathbf {Q}(\sqrt {D})$ has narrow class number one, and $A/F$ an abelian surface of $\operatorname {GL}_2$-type with everywhere good reduction. Assuming that $A$ is modular, we show that $A$ is either a $\mathbf {Q}$-surface or is a base change from $\mathbf {Q}$ of an abelian surface $B$ such that $\operatorname {End}_\mathbf {Q}(B) = \mathbf {Z}$, 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.## References

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## Additional Information

**Lassina Dembélé**- Affiliation: Department of Mathematics, University of Luxembourg, Esch-sur-Alzette L-4364, Luxembourg
- ORCID: 0000-0001-9001-5035
- Email: lassina.dembele@gmail.com
- Received by editor(s): April 5, 2020
- Received by editor(s) in revised form: April 6, 2020, August 2, 2021, and August 15, 2021
- Published electronically: December 3, 2021
- Additional Notes: The author was supported by EPSRC Grants EP/J002658/1 and EP/L025302/1, a Simons Collaboration Grant (550029) and by the Luxembourg National Research Fund PRIDE/GPS/12246620
- © Copyright 2021 American Mathematical Society
- Journal: Math. Comp.
**91**(2022), 1381-1403 - MSC (2020): Primary 11G10
- DOI: https://doi.org/10.1090/mcom/3692
- MathSciNet review: 4405499