THE EISENSTEIN IDEAL OF WEIGHT <i>k</i> AND RANKS OF HECKE ALGEBRAS

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