Estimation of genus of arithmetic curves and applications

Abstract

For any element gamma is an element of Gamma(0) (N) and a positive integer N, we find the genus of arithmetic curve [Gamma(1)(N), gamma Phi]\h*, where Phi = (0 -1 N 0) is the Fricke involution. We obtain that the genus of [Gamma(1)(N), gamma Phi]\h*, is zero if and only if 1 <= N <= 12 or N = 14, 15. As its applications, since the genus formula is independent of gamma, we determine the Hauptmoduln for the groups [Gamma(1) (N), Phi]of genus zero which will be used to generate appropriate ray class fields over imaginary quadratic fields, and show that the fixed point of gamma Phi in h is a Weierstrass point of for all but finitely many N, which is a direct generalization of Lehner-Newman's use of Schoeneberg's Theorem.