## Journal of the Mathematical Society of Japan

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

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

#### 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 ${F}_{1}(3/5,3/5,2/5,6/5)$.

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