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arXiv · 1804.06400

The Eisenstein ideal with squarefree level

Preston Wake·Carl Wang-Erickson
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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.

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Preston Wake, Carl Wang-Erickson. 2018-04-17. The Eisenstein ideal with squarefree level. https://doi.org/10.1016/j.aim.2020.107543

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