论文标题

$ x_0(n^2m)$的模块化单位和尖锐的除数类,$ n | 24 $和$ m $ squarefree

Modular units and cuspidal divisor classes on $X_0(n^2M)$ with $n|24$ and $M$ squarefree

论文作者

Wang, Liuquan, Yang, Yifan

论文摘要

对于一个正整数$ n $,令$ \ mathscr c(n)$为$ j_0(n)$的子组,由$ 0 $ 0 $ 0 $ 0 $和$ \ mathscr c(n)(\ Mathbb Q)的等价类别产生的类别产生Q $ - 理性亚组。也让$ \ mathscr c _ {\ mathbb q}(n)$是$ \ mathscr c(n)的子组(\ mathbb q)$由$ \ mathbb q $ - 理性cuspidal除数生成。我们证明,当某些整数$ n $划分$ 24 $和一些SquareFree Integer $ m $时,两组$ \ Mathscr c(n)(\ Mathbb Q)$和$ \ Mathscr C _ {\ Mathbb Q}(N)$相等。为了实现这一目标,我们表明,此类$ n $上$ x_0(n)$上的所有模块化单元是$η(mτ+k/h)$,$ mh^2 | n $和$ k \ in \ mathbb z $的功能的产品,并确定$ x_0(n)$ x_0(n)$ x_0(n)$ x_0(n)$ x_0(n)$ x_0的必要条件。

For a positive integer $N$, let $\mathscr C(N)$ be the subgroup of $J_0(N)$ generated by the equivalence classes of cuspidal divisors of degree $0$ and $\mathscr C(N)(\mathbb Q):=\mathscr C(N)\cap J_0(N)(\mathbb Q)$ be its $\mathbb Q$-rational subgroup. Let also $\mathscr C_{\mathbb Q}(N)$ be the subgroup of $\mathscr C(N)(\mathbb Q)$ generated by $\mathbb Q$-rational cuspidal divisors. We prove that when $N=n^2M$ for some integer $n$ dividing $24$ and some squarefree integer $M$, the two groups $\mathscr C(N)(\mathbb Q)$ and $\mathscr C_{\mathbb Q}(N)$ are equal. To achieve this, we show that all modular units on $X_0(N)$ on such $N$ are products of functions of the form $η(mτ+k/h)$, $mh^2|N$ and $k\in\mathbb Z$ and determine the necessary and sufficient conditions for products of such functions to be modular units on $X_0(N)$.

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