 
Summary: ARITHMETIC TORELLI MAPS FOR CUBIC SURFACES AND
THREEFOLDS
JEFFREY D. ACHTER
ABSTRACT. 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. This answers, away
from the prime 2, an old question of Deligne and a recent question of Kudla and
Rapoport. We further classify the MumfordTate groups of the abelian varieties
which arise, and give additional arithmetic applications.
1. INTRODUCTION
Consider a complex cubic surface. By associating to it either a K3 surface [DvGK05]
or a cubic threefold [ACT98, ACT02], one can construct a principally polarized
Hodge structure of level one. The Hodge structures which arise are parametrized
by B4, the complex 4ball, and in this way one can show that the moduli space SC
of complex cubic surfaces is uniformized by B4.
It turns out that the relevant arithmetic quotient of B4 is the set of complex points
of M, the moduli space of abelian fivefolds with action by Z[3] of signature (4, 1),
and there is a Torelli map C : SC MC. The main goal of the present paper is to
construct (Theorem 4.5) a Torelli morphism
S
