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

Abstract

For a positive integer 𝑁, let 𝑋 0 (𝑁) be the modular curve over 𝐐 and 𝐽 0 (𝑁) its Jacobian variety. We prove that the rational cuspidal subgroup of 𝐽 0 (𝑁) is equal to the rational cuspidal divisor class group of 𝑋 0 (𝑁) when 𝑁 = 𝑝 2 𝑀 for any prime 𝑝 and any squarefree integer 𝑀. To achieve this, we show that all modular units on 𝑋 0 (𝑁) can be written as products of certain functions 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) under a mild assumption.