Journal of the Mathematical Society of Japan

On the family of pentagonal curves of genus 6 and associated modular forms on the ball

Kenji KOIKE

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Abstract

In this article we study the inverse of the period map for the family F of complex algebraic curves of genus 6 equipped with an automorphism of order 5 having 5 fixed points. This is a family with 2 parameters, and is fibred over a Del Pezzo surface. Our period map is essentially same as the Schwarz map for the Appell hypergeometric differential equation F1(3/5,3/5,2/5,6/5).

This differential equation and the family F are studied by G. Shimura (1964), T. Terada (1983, 1985), P. Deligne and G. D. Mostow (1986) and T. Yamazaki and M. Yoshida (1984). Based on their results we give a representation of the inverse of the period map in terms of Riemann theta constants. This is the first variant of the work of H. Shiga (1981) and K. Matsumoto (1989, 2000) to the co-compact case.

Article information

Source
J. Math. Soc. Japan, Volume 55, Number 1 (2003), 165-196.

Dates
First available in Project Euclid: 5 December 2007

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1196890848

Digital Object Identifier
doi:10.2969/jmsj/1196890848

Mathematical Reviews number (MathSciNet)
MR1939191

Zentralblatt MATH identifier
1038.14010

Subjects
Primary: 14K25: Theta functions [See also 14H42]
Secondary: 11F55: Other groups and their modular and automorphic forms (several variables)

Keywords
theta function algebraic curve configuration space

Citation

KOIKE, Kenji. On the family of pentagonal curves of genus 6 and associated modular forms on the ball. J. Math. Soc. Japan 55 (2003), no. 1, 165--196. doi:10.2969/jmsj/1196890848. https://projecteuclid.org/euclid.jmsj/1196890848


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