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We study the rationality properties of the moduli space $\mathcal{A}_g$ ofprincipally polarised abelian $g$-folds over $\mathbb{Q}$ and apply the resultsto arithmetic questions. In particular we show that any principally polarisedabelian 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-Langconjecture we also show that this is not the case for abelian varieties ofdimension at least seven. About moduli spaces, we show that $\mathcal{A}_g$ isunirational 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 andZannier about the existence of abelian varieties over $\mathbb{Q}$ that are notisogenous to Jacobians.

Rationality and arithmetic of the moduli of abelian varieties