Arithmetic Torelli Maps for Cubic Surfaces and Threefolds

It has long been known that to a complex cubic surface or threefold one can canonically associate a principally polarized abelian variety. We give a construction which works for cubics over an arithmetic base, and in particular identifies the moduli space of cubic surfaces with an open substack of a certain moduli space of abelian varieties. This answers, away from the prime 2, an old question of Deligne and a recent question of Kudla and Rapoport.