# Arithmetic aspects of the Burkhardt quartic threefold

## TL;DR

This paper investigates the arithmetic properties of the Burkhardt quartic threefold, demonstrating its rationality over various fields, computing its zeta function over finite fields, and exploring its moduli interpretations and geometric structures.

## Contribution

It provides explicit models and interpretations of the Burkhardt quartic, including its rationality, zeta function, and moduli space connections, which were previously not fully understood.

## Key findings

- Proves the Burkhardt quartic is rational over fields with characteristic not 3.
- Computes the zeta function of the threefold over finite fields.
- Establishes a moduli interpretation via a universal genus 2 curve and describes geometric features like j-planes and Hesse pencils.

## Abstract

We show that the Burkhardt quartic threefold is rational over any field of characteristic distinct from 3. We compute its zeta function over finite fields. We realize one of its moduli interpretations explicitly by determining a model for the universal genus 2 curve over it, as a double cover of the projective line. We show that the j-planes in the Burkhardt quartic mark the order 3 subgroups on the Abelian varieties it parametrizes, and that the Hesse pencil on a j-plane gives rise to the universal curve as a discriminant of a cubic genus one cover.