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Rationality and arithmetic of the moduli of abelian varieties
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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.
