Journal of the Mathematical Society of Japan

A variant of Jacobi type formula for Picard curves

Keiji MATSUMOTO and Hironori SHIGA

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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 ) .

Article information

Source
J. Math. Soc. Japan, Volume 62, Number 1 (2010), 305-319.

Dates
First available in Project Euclid: 5 February 2010

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1265380432

Digital Object Identifier
doi:10.2969/jmsj/06210305

Mathematical Reviews number (MathSciNet)
MR2648224

Zentralblatt MATH identifier
1188.33022

Subjects
Primary: 33C65: Appell, Horn and Lauricella functions 14K25: Theta functions [See also 14H42]

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

Citation

MATSUMOTO, Keiji; SHIGA, Hironori. A variant of Jacobi type formula for Picard curves. J. Math. Soc. Japan 62 (2010), no. 1, 305--319. doi:10.2969/jmsj/06210305. https://projecteuclid.org/euclid.jmsj/1265380432


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References

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