The cuspidal class number formula for certain quotient curves of the modular curve X 0 ( M ) by Atkin-Lehner involutions

Abstract

We calculate the cuspidal class number of a certain quotient curve of the modular curve $X_0(M)$ with $M$ square-free. For each factor $r$ of $M$, let $w_r$ denote the Atkin-Lehner type involution of $X_0(M)$. Let $M_0$ be a divisor of $M$, and $W_0$ the subgroup of the automorphism group of $X_0(M)$ consisting of all $w_r$ with $r$ dividing $M_0$. Our object is the quotient of $X_0(M)$ by $W_0$. In this paper, we consider the case where $M$ is odd.