Rationality and arithmetic of the moduli of abelian varieties

We study the rationality properties of the moduli space Ag of principally polarised abelian g-folds over Q and apply the results to arithmetic questions. In particular, we show that any principally polarised abelian 3-fold over Fp may be lifted to an abelian variety over 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 Ag
is unirational over Q for g≤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 Q that are not isogenous to Jacobians.