Let $p$ and $\ell$ be primes such that $p > 3$ and $p \mid \ell-1$ and $k$ be an even integer. We will give a necessary and sufficient condition for the $Z_p$-rank of 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 $p$ to be greater than $1$ in terms of vanishing of the cup products of certain global Galois cohomology classes. We will begin with a brief review of modular forms.
