The classical Jacobi formula for the elliptic integrals shows a relation between Jacobi’s theta constants and periods of elliptic curves \(E(\lambda): w^2=z(z-1)(z-\lambda)\). In other words, it expresses the modular form \(\vartheta^{4} _{00}(\tau)\) with respect to the principal congruence subgroup \(\Gamma (2)\) of \(\text{PSL}(2,\mathbb Z)\) in terms of the Gauss hypergeometric function \(F(1/2, 1/2, 1; 1-\lambda)\). This is done by the inverse of the period map for the family of elliptic curves \(E(\lambda)\).
A similar formula is given for the family of (genus 3) Picard curves \(C(\lambda_1,\lambda_2): w^3=z(z-1)(z-\lambda_1)(z-\lambda_2)\):
The inverse of the period map for \(C(\lambda_1,\lambda_2)\) establishes an expression of the modular form \(\vartheta_0^3(u,v)\) (defined on a two dimensional complex ball \(\mathcal{D}=\{ 2\operatorname{Re}v+|u|^2<0\}\), that can be realized as a Shimura variety in the Siegel upper half space of degree 3) in terms of the Appell hypergeometric function \(F_1(1/3, 1/3, 1/3, 1; 1-\lambda_1,1-\lambda_2)\).