Abstract
Abstract
Let p and
$\ell $
be primes such that
$p> 3$
and
$p \mid \ell -1$
and k be an even integer. We use deformation theory of pseudo-representations to study the completion of the Hecke algebra acting on the space of cuspidal modular forms of weight k and level
$\Gamma _0(\ell )$
at the maximal Eisenstein ideal containing p. We give a necessary and sufficient condition for the
$\mathbb {Z}_p$
-rank of this Hecke algebra to be greater than
$1$
in terms of vanishing of the cup products of certain global Galois cohomology classes. We also recover some of the results proven by Wake and Wang-Erickson for
$k=2$
using our methods. In addition, we prove some
$R=\mathbb {T}$
theorems under certain hypotheses.
Publisher
Cambridge University Press (CUP)
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