Rationality and arithmetic of the moduli of abelian varieties

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 3-fold 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 7. Concerning moduli spaces, we show that ${\mathcal{A}}_g$ is unirational over $\mathbb{Q}$ for $g\le 5$ 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.