A variant of Jacobi type formula for Picard curves

概要

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The classical Jacobi formula for the elliptic integrals (Gesammelte Werke I, p. 235) shows a relation between Jacobi theta constants and periods of ellptic curves E(λ):w2=z(z−1)(z−λ). In other words, this formula says that the modular form ϑ400(τ) with respect to the principal congruence subgroup Γ(2) of PSL(2,Z) has an expression by the Gauss hypergeometric function F(1⁄2,1⁄2,1;1−λ) via the inverse of the period map for the family of elliptic curves E(λ) (see Theorem 1.1). In this article we show a variant of this formula for the family of Picard curves C(λ1,λ2):w3=z(z−1)(z−λ1)(z−λ2), those are of genus three with two complex parameters. Our result is a two dimensional analogy of this context. The inverse of the period map for C(λ1,λ2) is established in [S] and our modular form ϑ03(u,v) (for the definition, see (2.7)) is defined on a two dimensional complex ball D={2Rev+|u|2<0}, that can be realized as a Shimura variety in the Siegel upper half space of degree 3 by a modular embedding. Our main theorem says that our theta constant is expressed in terms of the Appell hypergeometric function F1(1⁄3,1⁄3,1⁄3,1;1−λ1,1−λ2).