5 problems

Let $U^p$ be a compact open subgroup away from $p$, let $\mathcal{O}$ be the coefficient ring, let $\mathbb{T}^S(U^p)$ be the Hecke algebra, and let $q_0$ be the distinguished homo…

Local-global compatibility conjecture. The action of $R_p^{\operatorname{loc}}$ on

Taylor–Wiles-level lifting conjecture. If $\overline{\rho}_{\mathfrak{m}}$ is absolutely irreducible, there exists a continuous lifting

Let $Q$ be an auxiliary Taylor–Wiles set, let $\overline{\rho}_{\mathfrak m}$ be the residual representation, let $\mu$ be the specified character, and let $r_{{\mathfrak m},Q}$ be…

Let $Q$ be an auxiliary Taylor–Wiles set and assume the hypotheses of the stated proposition, with ${\mathfrak m}_{Q,1}$ and $F^\bullet_{{\mathfrak m}_{Q,1}}$ defined there. Taylor…