The Eisenstein ideal of weight $k$ and ranks of Hecke algebras

Abstract

Let p and ℓ be primes such that p > 3 and p | ℓ − 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 Γ0(ℓ) at the maximal Eisenstein ideal containing p. We give a necessary and sufficient condition for the Zp-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 = T theorems under certain hypothesis.