Integral Galois representation conjecture for non-Eisenstein Hecke ideals

Let $F$ be a number field and let $\mathbb T^S(K,\lambda)$ be the faithful Hecke algebra acting on the cohomology of the associated locally symmetric space. Let $frak m$ be a non-Eisenstein maximal ideal, and let $\mathbb T^S(K,\lambda)^{\widehat{\ }}_{\mathfrak m}$ denote its completion.

Integral Galois representation conjecture. There exists a unique continuous representation

$$\rho_{\mathfrak m}:\Gamma_F\longrightarrow\operatorname{GL}_n\left(\mathbb T^S(K,\lambda)^{\widehat{\ }}_{\mathfrak m}\right)$$

that is unramified at every finite place $v\notin S$ and whose Frobenius characteristic polynomial is the image of

$$X^n-T_{v,1}X^{n-1}+\dots+(-1)^iq_v^{i(i-1)/2}T_{v,i}X^{n-i}+\dots+(-1)^nq_v^{n(n-1)/2}T_{v,n}$$

in the completed Hecke algebra.

This conjecture lifts the residual Galois representation attached to a non-Eisenstein maximal ideal to the completed Hecke algebra. It is a key input for integral local-global compatibility and Taylor–Wiles–type arguments.