Frank Calegari, Matthew Emerton
Published 2003-11-21Version 1
Using the modularity technique of Wiles, we study the Hecke algebra of weight 2 and prime level N localized at the Eisenstein primes. On the way, we recover some results of Mazur ("Modular Curves and the Eisenstein Ideal") from a deformation theoretic point of view. Combining some of our results with a theorem of Merel, we obtain new information about the p-part of the class groups of Q(N^(1/p)), where p and N are prime, and N = 1 mod p.
A Classification of order $4$ class groups of $\mathbb{Q}{(\sqrt{n^2+1})}$
$\ell$-Torsion in Class Groups via Dirichlet $L$-functions
The Eisenstein ideal of weight $k$ and ranks of Hecke algebras
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