Rationality and arithmetic of the moduli of abelian varieties
Daniel Loughran, Gregory Sankaran
TL;DR
The paper investigates the arithmetic and birational geometry of the moduli space $\mathcal{A}_g$ of principally polarised abelian varieties over $\mathbb{Q}$, establishing lifting properties from $\mathbb{F}_p$ to $\mathbb{Q}$ in low dimensions and delineating rationality properties of $\mathcal{A}_g$ across dimensions. It proves that $\mathcal{A}_g$ is unirational over $\mathbb{Q}$ for $g\le 5$ and that $\mathcal{A}_3$ is stably rational, with $\mathcal{A}_2$ rational; for $g\ge 7$ the Bombieri–Lang conjecture implies that many reductions mod $p$ cannot lift to characteristic zero, revealing a sharp dimension-dependent behavior. The work also develops a weak approximation framework for the moduli stack $\mathscr{A}_3$, enabling arithmetic applications such as the existence of abelian varieties over $\mathbb{Q}$ not isogenous to Jacobians in low dimensions. Collectively, these results connect moduli geometry with arithmetic lifting questions and provide unconditional statements (via Prym constructions) in dimension four and five, while outlining open questions for higher dimensions and stronger approximation properties.
Abstract
We study the rationality properties of the moduli space $\mathcal{A}_g$ of principally polarised abelian $g$-folds over $\mathbb{Q}$ and apply the results to arithmetic questions. In particular we show that any principally polarised abelian threefold over $\mathbb{F}_p$ may be lifted to an abelian variety over $\mathbb{Q}$. This is a phenomenon of low dimension: assuming the Bombieri-Lang conjecture we also show that this is not the case for abelian varieties of dimension at least seven. About moduli spaces, we show that $\mathcal{A}_g$ is unirational over $\mathbb{Q}$ for $g \leq 5$ and stably rational for $g=3$. This also allows us to make unconditional one of the results of Masser and Zannier about the existence of abelian varieties over $\mathbb{Q}$ that are not isogenous to Jacobians.