Quantitative level lowering for Galois representations

We use Galois cohomology methods to produce optimal mod $p^d$ level lowering congruences to a $p$-adic Galois representation that we construct as a well chosen lift of a given residual mod $p$ representation. Using our explicit Galois cohomology methods, we construct for a reductive group $G$ and a given residual representation $\bar{\rho }:{\mathrm{\Gamma }}_F\to G(k)$, ramified at a finite set of primes $S$, in favorable conditions that we identify, a finite set of lifts $\rho$, $\{{\rho }^q\}$ of $\bar{\rho }$ to $G(W(k))$ with the following properties: $\rho :{\mathrm{\Gamma }}_F\to G(W(k))$ is ramified precisely at $S\cup Q$, with $Q$ a finite set of primes disjoint from $S$. For $q\in Q$, $\rho^q:G_F \to G(W(k))$isunramifiedoutside$S \cup Q \backslash {q}$and$\rho$and$\rho^q$arecongruentmod$p^d$if$\rho$mod$p^d$isunramifiedat$q$. Furthermore, the Galois representations $\{{\rho }^q\}$ are “independent”.