Genus two curves with everywhere good reduction over quadratic fields

TL;DR

This paper constructs infinite families of genus 2 curves over quadratic fields with everywhere good reduction, providing new examples of such curves with absolutely simple Jacobians, advancing understanding in arithmetic geometry.

Contribution

It introduces the first infinite sequences of genus 2 curves with everywhere good reduction over quadratic fields, including both real and complex cases, with Jacobians that are absolutely simple.

Findings

Constructed infinite sequences of genus 2 curves over quadratic fields.

Demonstrated Jacobians are absolutely simple abelian varieties.

Provided examples over both real and complex quadratic fields.

Abstract

We address the question of existence of absolutely simple abelian varieties of dimension 2 with everywhere good reduction over quadratic fields. The emphasis will be given to the construction of pairs $(K,C)$, where $K$ is a quadratic number field and $C$ is a genus $2$ curve with everywhere good reduction over $K$. We provide the first infinite sequence of pairs $(K,C)$ where $K$ is a real (complex) quadratic field and $C$ has everywhere good reduction over $K$. Moreover, we show that the Jacobian of $C$ is an absolutely simple abelian variety.