AI 中文总结
该研究针对任意整数\(N\)的模曲线\(X_1(N)\)，给出有理尖点类群阶数的显式公式，证明依赖斯特伦的结果及相关行列式计算，还定义了高权类似物并推测其阶数与特定行列式有关。
AI 中文摘要
我们给出了任意整数\(N\)时，模曲线\(X_1(N)\)的有理尖点类群阶数的显式公式。证明依赖于斯特伦关于\(X_1(N)\)上模单位群的结果，且需要计算一个涉及第二类伯努利多项式的行列式。我们还定义了尖点类群的高权类似物，并推测其阶数与用更高次伯努利多项式定义的类似行列式有关。
英文摘要
We give an explicit formula for the order of the rational cuspidal class group of the modular curve $X_1(N)$ for an arbitrary integer $N$. The proof relies on results of Streng on the group of modular units on $X_1(N)$, and requires computing a certain determinant involving the second Bernoulli polynomial. We also define a higher weight analogue of the cuspidal class group and speculate that its order is related to a similar determinant defined using a higher degree Bernoulli polynomial.
