We apply the methods of Fukaya, Kato and Sharifi to refine Mazur's study of the Eisenstein ideal. Given prime numbers N and $p\ge 5$ such that $p|\varphi (N)$, we study the quotient of the cohomology group of modular curve $X_0(N)$ by the square of the Eisenstein ideal. We study two invariants $b,c$ attached to this quotient and compute c. We propose a conjecture about the invariant b which relates the structure of the ray class group of conductor N to the modular symbols of $X_0(N)$. Assuming this conjecture, We compute the invariant b.

中文翻译：

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走向Sharifi的猜想

我们运用深谷（Fukaya），加藤（Kato）和沙里菲（Sharifi）的方法来完善马祖尔对爱森斯坦理想的研究。给定素数N和$p\ge 5$ 这样 $p|\varphi （ñ）$，我们研究了模块化曲线的同调群的商 $X_0（ñ）$爱森斯坦理想主义的广场。我们研究两个不变量$b，C$附加到该商并计算c。我们提出一个关于不变量b的猜想，该不变量b将导体N的射线类别组的结构与的模块化符号相关联$X_0（ñ）$。假设这个猜想，我们计算不变量b。