# A Prym variety with everywhere good reduction over $\mathbb{Q}(\sqrt{61})$

## TL;DR

This paper constructs a specific modular abelian surface over () with good reduction everywhere over () that lacks a principal polarization over the same field, highlighting unique reduction properties.

## Contribution

It provides an explicit equation for a modular abelian surface with everywhere good reduction over () that does not admit a principal polarization, a novel example in the study of abelian varieties.

## Key findings

- Explicit equation for the abelian surface over ()
- Demonstrates the existence of a surface with good reduction everywhere without a principal polarization
- Highlights unique properties of abelian surfaces over quadratic fields.

## Abstract

We compute an equation for a modular abelian surface $A$ that has everywhere good reduction over the quadratic field $K = \mathbb{Q}(\sqrt{61})$ and that does not admit a principal polarization over $K$.