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≥5 such that p|ϕ(N), we study the quotient of the cohomology group of modular curve X0(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 X0(N). Assuming this conjecture, we compute the invariant b.
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