Twists of the Burkhardt Quartic Threefold

TL;DR

This paper investigates twists of the Burkhardt quartic threefold over various fields, identifying Galois cohomological obstructions to their rationality and their relation to moduli spaces of abelian varieties.

Contribution

It introduces a Galois-cohomological framework to analyze twists of the Burkhardt quartic and their connection to abelian surfaces and rational points.

Findings

All twists admit quartic models in projective four-space.

A Galois cohomological obstruction determines when a twist is birational to a moduli space.

Some twists have no rational points over certain fields.

Abstract

We study twists of the Burkhardt quartic threefold over non-algebraically closed base fields of characteristic different from 2,3,5. We show they all admit quartic models in projective four-space. We identify a Galois-cohomological obstruction that measures if a given twist is birational to a moduli space of abelian varieties. This obstruction has implications for the rational points on these varieties. As a result, we see that all possible 3-level structures can be realized by abelian surfaces, whereas Kummer 3-level structures that group-theoretically may be admissible, may not be realizable over certain base fields. We give an example of a Burkhardt quartic over a bivariate function field whose desingularization has no rational points at all. Our methods are based on the representation theory of Sp(4,3), Galois cohomology, and the classical algebraic geometry of the Burkhardt quartic.