The Cuspidal Class Number Formula for the Modular CurvesX0(M) withMSquare-Free

✍ Scribed by Toshikazu Takagi

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 413 KB
• Volume: 193
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1996.6993

✦ Synopsis

CUSPIDAL CLASS NUMBER FORMULA 181 the group ring. In Section 3, we prove that all modular units on the Ž . modular curve X M can be written as products of the functions h and 0 rational numbers. In Section 4, we determine a necessary and sufficient condition under which a product of the functions h is a modular unit on Ž . the curve X M . Here, we use the properties of the generalized Siegel 0 functions. Newman used the property of Dedekind sums and obtained a sufficient condition. In Section 5, we calculate the cuspidal class number. Our main theorem is Theorem 5.1. In the calculation, we use the ring structure of R. The image of the divisors of the modular units is an ideal of the ring R and is an analogue of the Stickelberger ideal in the theory of cyclotomic fields. In Section 6, we determine the p-Sylow group of the cuspidal divisor class group for p / 2, 3, and, under certain conditions on M, the 3-Sylow group.

In this paper, we denote by Z, Q, and C the ring of rational integers, the field of rational numbers, and the field of complex numbers, respectively. 1. TRANSFORMATION FORMULAS FOR SIEGEL FUNCTIONS ' Let O O be the order defined by O O s Ý Z r . Let n, m be positive r g T ' integers such that m is a divisor of M. Put I s n m O O. Then I is an ideal of O O. We assume that N s nm / 1. In this section we consider the Siegel functions with respect to the ideal I. I 1