On the Construction of Class Fields by Picard Modular Forms
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The goal of this article is to construct modular functions living on the complex ball of dimension two such that the values in special points — similar to the elliptic modular function — generate class fields. For this purpose we are well prepared by the papers  and . The first one classifies the moduli space of abelian 3-folds with a multiplication by ℚ(i) of type (2, 1) as projective surface. Via Jacobians we connect this Shimura surface with the moduli space of a family of curves of Shimura equation type. Thus we are able to continue the construction of the inverse period map of the family by theta constants given in . Knowing the action of the modular group we reach a modular function j by modular forms with respect to the congruence subgroup of level (1 + i) of the full Picard modular group of Gauß numbers. If τ is the period of a (Jacobian of a) curve with complex multiplication the corresponding moduli field is generated over the rational numbers by j(τ). Hence the values in CM points of this function generate abelian extensions of the associated reflex field.
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