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摘要
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 v(00)(4)(tau) with respect to the principal congruence subgroup Gamma(2) of PSL(2, 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)) : omega(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 (3)(0)(u, nu) (for the definition, see (2.7)) is defined oil a two dimensional complex ball D = {2Rev + vertical bar u vertical bar(2) < 0}, that call be realized as a Shimura variety in the Siegel tipper 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)).
- 出版日期2010-1
