The rational cuspidal subgroup of J0(p2M)$J_0(p^2M)$ with M squarefree期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

The rational cuspidal subgroup of J0(p2M)$J_0(p^2M)$ with M squarefree

Authors:	Jia-Wei Guo Yifan Yang Hwajong Yoo Myungjun Yu

Affiliation:	1. Department of Mathematics, National Taiwan University, Taipei, Taiwan;2. Department of Mathematics, National Taiwan University, Taipei, Taiwan National Center for Theoretical Science, Taipei, Taiwan;3. College of Liberal Studies and Research Institute of Mathematics, Seoul National University, Seoul, South Korea;4. Department of Mathematics, Yonsei University, Seoul, South Korea

Abstract:	For a positive integer N, let $X_{0} (N) $X_0(N)$$ be the modular curve over $Q $\mathbf {Q}$$ and $J_{0} (N) $J_0(N)$$ its Jacobian variety. We prove that the rational cuspidal subgroup of $J_{0} (N) $J_0(N)$$ is equal to the rational cuspidal divisor class group of $X_{0} (N) $X_0(N)$$ when $N = p^{2} M $N=p^2M$$ for any prime p and any squarefree integer M. To achieve this, we show that all modular units on $X_{0} (N) $X_0(N)$$ can be written as products of certain functions $F_{m, h} $F_{m, h}$$ , which are constructed from generalized Dedekind eta functions. Also, we determine the necessary and sufficient conditions for such products to be modular units on $X_{0} (N) $X_0(N)$$ under a mild assumption.

Keywords:	cuspidal subgroup modular units rational cuspidal subgroup