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 $\barρ: Γ_F \to G(k)$, ramified at a finite set of primes $S$, in favorable conditions that we identify, a finite set of lifts $ρ$, $\{ρ^q\}$ of $\barρ$ to $G(W(k))$ with the following properties: $ρ: Γ_F \to G(W(k))$ is ramified precisely at $S \cup Q$, with $Q$ a finite set of primes disjoint from $S$. For $q \in Q$, $ρ^q:G_F \to G(W(k))$ is unramified outside $S \cup Q \backslash \{q\}$ and $ρ$ and $ρ^q$ are congruent mod $p^d$ if $ρ$ mod $p^d$ is unramified at $q$. Furthermore, the Galois representations $\{ρ^q\}$ are "independent".

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