A refined study of Mazur's Eisenstein theory

Abstract

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 $p\mid \phi (N)$, 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 $b$.

major revision to improve exposition, change the title and rewrite the introduction