Theta Constants Associated to Cubic Three Folds

Every elliptic curve is isomorphic to the double covering of the projective line P branching at four points. The set of the isomorphism classes of elliptic curves with marking of branching points can be identified with the configuration space M4pts of ordered four points on P. By considering their periods, one obtains a morphism per from M4pts to the set Mct of the isomorphism classes of complex torus with marking of 2-torsion points. The set Mct can be identified with the quotient space H/Γ(2) of the upper half plane H by the principal congruence subgroup Γ(2) of level 2 in SL(2,Z). By the classical theory of elliptic functions, the map per : M4pts → Mct is an isomorphism. The inverse of per can be described in terms of celebrated Jacobi’s theta constants. Many mathematicians have tried to find moduli spaces of suitable algebraic varieties which can be uniformized by some symmetric space. E. Picard was the first person who found such moduli space which is two dimensional. He studied a family of Picard curves, which are cyclic triple coverings of P branching at five points. The periods of a curve determine an element of the 2-dimensional complex ball B2 embedded in the Siegel upper-half space H3 of degree 3. This correspondence gives a uniformization of the moduli space of the Picard curves by B2. The inverse Θ of the period map was expressed in terms of special values of theta functions for the Jacobians of Picard curves. Shiga then found the representation of the map Θ in terms of theta constants, which are automorphic forms on B2 with respect to the monodromy group. Inspired by Picard’s results, the moduli spaces of the cyclic coverings of P branching at (n+3)-points uniformized by the n-dimensional complex balls were classified by Terada ([T]), Deligne and Mostow ([DM]). One specific three dimensional moduli spaces listed in [DM] was studied in [Ma] similarly to Shiga: the inverse of the period map for a family of cyclic triple coverings of P branching at six points was expressed in terms of theta constants.