Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Eisenstein ideal with squarefree level   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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The categories of this paper are: alg-geom
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The categories of this paper are: solv-int
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The categories of this paper are: bayes-an
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The categories of this paper are: supr-con
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The categories of this paper are: atom-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-th
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The categories of this paper are: funct-an
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The categories of this paper are: dg-ga
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: cs
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The categories of this paper are: physics
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The categories of this paper are: nlin
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The categories of this paper are: mtrl-th