The cuspidal class number formula for certain quotient curves of the modular curve \(X_0(M)\) by Atkin-Lehner involutions. (English) Zbl 1273.11095

Takagi, Toshikazu

J. Math. Soc. Japan 62, No. 1, 13-47 (2010).

Summary: 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.

MSC:

11G18	Arithmetic aspects of modular and Shimura varieties
11G16	Elliptic and modular units
14G35	Modular and Shimura varieties

Keywords:

modular curve; modular unit; cuspidal class number

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