In Wiles' celebrated paper where any semi-stable elliptic curve $E$ over ${\Bbb Q}$ is modular, Theorem $0.3.$ therein assumes that either $E$ is good or multiplicative reduction at $3$. This condition seems to restrict the Hecke algebra ${\Bbb T}_m(N)$ to cases where either
- $p \nmid N$
or
- ${\Bbb T}_m(N)$ is an ordinary Hecke algebra, i.e., $U_p \notin {\frak m}$.
In either cases in the above, Haruzo Hida had already established De Shalit's lemma which asserts that Hecke algebra ${\Bbb T}_{\frak m}(Nq_1 \cdots q_n)$ is free over ${\Bbb Z}_p[\Delta]$ with $\Delta$ generated by Diamond operators which are generators of $({\Bbb Z}/q)^{\times}$.
Q. Can Hida's result substitute for De Shalit's result which includes $p \mid N$ and non-ordinary case?
