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 $\mathcal{F}$ of complex algebraic curves of genus 6 equipped with an automorphism of order 5. This is a family with 2 parameters, and is fibred over a certain type of Del Pezzo surace. The period satisfies the hypergeometric differential equation for Appell's $F_1({3/5},{3/5},{2/5},{6/5})$ of two variables after a certain normalization of the variable parameter. This differential equation and the family $\mathcal{F}$ are studied by G. Shimura (1964), T. Terada (1983, 1985), P. Deligne - G.D. Mostow (1986) and T. Yamazaki- M. Yoshida(1984). Recently M. Yoshida presented a new approch using the concept of configration space. Based on their results we show the 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.