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January, 2010 A variant of Jacobi type formula for Picard curves
Keiji MATSUMOTO, Hironori SHIGA
J. Math. Soc. Japan 62(1): 305-319 (January, 2010). DOI: 10.2969/jmsj/06210305

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 ( λ ) : w 2 = z ( z - 1 ) ( z - λ ) . In other words, this formula says that the modular form ϑ 00 4 ( 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 ( λ 1 , λ 2 ) : w 3 = z ( z - 1 ) ( z - λ 1 ) ( z - λ 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 ( λ 1 , λ 2 ) is established in [S] and our modular form ϑ 0 3 ( u , v ) (for the definition, see (2.7)) is defined on a two dimensional complex ball D = { 2 Re v + | 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 F 1 ( 1 / 3 , 1 / 3 , 1 / 3 , 1 ; 1 - λ 1 , 1 - λ 2 ) .

Citation

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Keiji MATSUMOTO. Hironori SHIGA. "A variant of Jacobi type formula for Picard curves." J. Math. Soc. Japan 62 (1) 305 - 319, January, 2010. https://doi.org/10.2969/jmsj/06210305

Information

Published: January, 2010
First available in Project Euclid: 5 February 2010

zbMATH: 1188.33022
MathSciNet: MR2648224
Digital Object Identifier: 10.2969/jmsj/06210305

Subjects:
Primary: 14K25 , 33C65

Keywords: Appell’s hypergeometric function , period integral , Picard modular form , theta function

Rights: Copyright © 2010 Mathematical Society of Japan

Vol.62 • No. 1 • January, 2010
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