Rationality of twists of the Siegel modular variety of genus $2$ and level $3$

TL;DR

This paper proves that certain moduli spaces of abelian surfaces with specified Galois representations are not rational over the rational numbers, despite being rational over the complex numbers and unirational over the rationals.

Contribution

It establishes the non-rationality over  of moduli spaces associated with surjective Galois representations for genus 2 Siegel modular varieties with level 3.

Findings

The moduli space is not rational over  for surjective -Galois representations.

The space is rational over  and unirational over , but not rational.

The degree of the unirational map over  is 6.

Abstract

Let $\overline{\rho}: G_{\mathbf{Q}} \rightarrow {\rm GSp}_4(\mathbf{F}_3)$ be a continuous Galois representation with cyclotomic similitude character -- or, what turns out to be equivalent, the Galois representation associated to the $3$-torsion of a principally polarized abelian surface $A/\mathbf{Q}$. We prove that the moduli space $\mathcal{A}_2(\overline{\rho})$ of principally polarized abelian surfaces $B/\mathbf{Q}$ admitting a symplectic isomorphism $B[3] \simeq \overline{\rho}$ of Galois representations is never rational over $\mathbf{Q}$ when $\overline{\rho}$ is surjective, even though it is both rational over $\mathbf{C}$ and unirational over $\mathbf{Q}$ via a map of degree $6$.