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

Jia‐Wei Guo, Yifan Yang, Hwajong Yoo and Myungjun Yu

Mathematische Nachrichten, 2023, vol. 296, issue 10, 4634-4655

Abstract: For a positive integer N, let X0(N)$X_0(N)$ be the modular curve over Q$\mathbf {Q}$ and J0(N)$J_0(N)$ its Jacobian variety. We prove that the rational cuspidal subgroup of J0(N)$J_0(N)$ is equal to the rational cuspidal divisor class group of X0(N)$X_0(N)$ when N=p2M$N=p^2M$ for any prime p and any squarefree integer M. To achieve this, we show that all modular units on X0(N)$X_0(N)$ can be written as products of certain functions Fm,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 X0(N)$X_0(N)$ under a mild assumption.

Date: 2023
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https://doi.org/10.1002/mana.202200177

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