A variant of Jacobi type formula for Picard curves

Abstract

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 $E(\lambda ):w^2=z(z-1)(z-\lambda )$. In other words, this formula says that the modular form ${\vartheta }_{00}^4(\tau )$ with respect to the principal congruence subgroup $\mathrm{\Gamma }(2)$ of $\mathit{PSL}(2,\mathit{Z})$ has an expression by the Gauss hypergeometric function $F(1/2,1/2,1;1-\lambda )$ via the inverse of the period map for the family of elliptic curves $E(\lambda )$ (see Theorem 1.1). In this article we show a variant of this formula for the family of Picard curves $C({\lambda }_1,{\lambda }_2):w^3=z(z-1)(z-{\lambda }_1)(z-{\lambda }_2)$, 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 $C({\lambda }_1,{\lambda }_2)$ is established in [S] and our modular form ${\vartheta }_0^3(u,v)$ (for the definition, see (2.7)) is defined on a two dimensional complex ball , 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 $F_1(1/3,1/3,1/3,1;1-{\lambda }_1,1-{\lambda }_2)$.