Moduli of Abelian Varieties with Higher Degree Polarizations

Fix integers g, d ≥ 1 and let A _g,d denote the moduli stack associating to a scheme T the groupoid of pairs (A, λ), where A is an abelian scheme over T and λ : A → A^t is a polarization of degree d. Recall that this means that the kernel of λ is a finite flat group scheme over T of rank d², and that fppf locally on T there exists an ample line bundle L on A such that the map

$${\rm{\lambda }}_L :A \to A^t ,a \to \left[ {t_a^* L \otimes L^{ - 1} } \right]$$

is equal to λ. In this case if f : A → T is the structural morphism then f_*L is locally free of rank d on T and its formation commutes with arbitrary base change on T (see for example [36, I, §1] for a summary of basic properties of ample line bundles on abelian varieties).