The Eisenstein ideal with squarefree level

TL;DR

This paper investigates the structure of Eisenstein ideals in the $p$-adic Hecke algebra at squarefree levels, revealing non-principality and non-Gorenstein properties, and confirming a conjecture of Ribet.

Contribution

It introduces pseudodeformation theory to analyze Eisenstein ideals at squarefree levels, showing they are local complete intersections and identifying cases where classical properties fail.

Findings

Eisenstein part of the $p$-adic Hecke algebra is a local complete intersection.

Eisenstein ideal can be non-principal.

Cuspidal quotient of the Hecke algebra is not Gorenstein.

Abstract

We use pseudodeformation theory to study the analogue of Mazur's Eisenstein ideal with certain squarefree levels. Given a prime number $p>3$ and a squarefree number $N$ satisfying certain conditions, we study the Eisenstein part of the $p$-adic Hecke algebra for $\Gamma_0(N)$, and show that it is a local complete intersection and isomorphic to a pseudodeformation ring. We also show that in certain cases, the Eisenstein ideal is not principal and that the cuspidal quotient of the Hecke algebra is not Gorenstein. As a corollary, we prove that "multiplicity one" fails for the modular Jacobian in these cases. In a particular case, this proves a conjecture of Ribet.