On the curve \(X(9)\) (Q1273312)

The paper under review studies the modular curve \(X(9)\) which is the compactification of the factor \(\mathbb{H}/\Gamma(9)\) of the moduli space by the level group \(\Gamma(9)\). It is shown, that this curve has genus 10 and 36 cusps. It is given by the complete intersection of two cubic hypersurfaces given by the equations \[ -\varphi_1 \varphi^2_3+ \varphi^2_1 \varphi_4+ \varphi_3 \varphi^2_4 =0,\quad- \varphi^3_2+ \varphi^2_1\varphi_3-\varphi_1 \varphi^2_4+ \varphi^2_3 \varphi_4=0, \] where \(\varphi_l\) are given as \(q\)-series (or \(\theta\)-constants with characteristics), \[ \varphi_l= \sum^\infty_{n= -\infty} (-1)^nq^{k(n+ (2l-1)/2k)^2}, \] where \(k=9\) for the considered case and \(q=\exp \{\pi i\tau\}\). The basis for holomorphic differentials for the curve \(X(9)\) is shown to be of the form \(\eta^2(\tau) \varphi_l \varphi_m\), \(1\leq l\leq 4\), \(1\leq m\leq 4\), \(l\leq m\), where \[ \eta= q^{1\over 12} \sum^\infty_{n=-\infty}(-1)^n q^{3n^2+n}. \] The functions \(\varphi_l\) and \(\eta\) were used by Klein and Hurwitz to parametrize \(X(7)\) as \(\varphi^3_1 \varphi_3+ \varphi^3_3 \varphi_2-\varphi^3_2\varphi_1=0\) with holomorphic differentials of the form \(\eta^3 \varphi_l\), \(l=1,2,3\) and a similar result for \(X(9)\) is obtained in the paper. The result is an improvement of the one given earlier by \textit{H. M. Farkas}, \textit{Y. Kopeliovich} and \textit{I. Kra} [Commun. Anal. Geom. 4, No. 2, 207-259 (1996; Zbl 0860.11022)], where \(X(9)\) is represented as intersection of five hypersurfaces.