The rational cuspidal subgroup of J₀(p²M) with M squarefree

2021/09/01 by Guo, Jia-Wei, Yang, Yifan, Yoo, Hwajong +1
#11G18 #14G05 #14G35 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2109.00174

Abstract

For a positive integer N, let \mathscrC_N(ℚ) be the rational cuspidal subgroup of J₀(N) and \mathscrC(N) be the rational cuspidal divisor class group of X₀(N), which are both subgroups of the rational torsion subgroup of J₀(N). We prove that two groups \mathscrC_N(ℚ) and \mathscrC(N) are equal when N=p²M for any prime p and any squarefree integer M. To achieve this we show that all modular units on X₀(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 X₀(N) under a mild assumption.

Related

• Modular units and cuspidal divisor classes on X₀(n²M) with n|24 and M squarefree
• The rational cuspidal subgroup of J₀(N)
• The rational cuspidal divisor class group of X₀(N)
• The rational torsion subgroup of J₀(N)
• Modular unit and cuspidal divisor class groups of X₁(N)