A refined study of Mazur's Eisenstein theory

TL;DR

This paper refines Mazur's Eisenstein ideal study by applying advanced methods to analyze specific invariants related to modular curves, conjecturing a link to ray class groups, and computing these invariants under certain assumptions.

Contribution

It introduces a refined analysis of the Eisenstein ideal using Fukaya, Kato, and Sharifi's methods, proposing a conjecture connecting invariants to ray class groups, and computes these invariants assuming the conjecture.

Findings

Computed the invariant c of the Eisenstein ideal quotient.

Proposed a conjecture relating invariant b to ray class groups.

Under the conjecture, explicitly calculated the invariant b.

Abstract

We apply the methods of Fukaya, Kato and Sharifi to refine Mazur's study of the Eisenstein ideal. Given prime numbers $N$ and $p\geq 5$ such that $p\mid \varphi(N)$, 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 $b$.