# Rational points in the moduli space of genus two

## TL;DR

This paper constructs a comprehensive database of genus 2 curves over rationals with various height bounds, analyzes the distribution of rational points in their moduli space, and discusses related open problems.

## Contribution

It provides the first extensive database of genus 2 curves with detailed invariants and automorphisms, enabling new insights into rational points in moduli space.

## Key findings

- Database includes all genus 2 curves with specified height bounds.
- Distribution analysis of rational points in moduli space is presented.
- Open problems related to rational points and moduli are discussed.

## Abstract

We build a database of genus 2 curves defined over $\mathbb Q$ which contains all curves with minimal absolute height $h \leq 5$, all curves with moduli height $\mathfrak h \leq 20$, and all curves with extra automorphisms in standard form $y^2=f(x^2)$ defined over $\mathbb Q$ with height $h \leq 101$. For each isomorphism class in the database, an equation over its minimal field of definition is provided, the automorphism group of the curve, Clebsch and Igusa invariants. The distribution of rational points in the moduli space $\mathcal M_2$ for which the field of moduli is a field of definition is discussed and some open problems are presented.