Quantitative level lowering for Galois representations

Abstract

We use Galois cohomology methods to produce optimal mod pd 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, for F a number field, ΓF its absolute Galois group and G a reductive group, k a finite field and a suitable representation ρ¯:normalΓF→Gfalse(kfalse), ramified at a finite set of primes S, we construct under favorable conditions lifts ρ, {ρq} of trueρ¯ to G(W(k)) for q∈Q with Q a finite set of places of F. The lifts {ρq} have the following properties: ρ:normalΓF→Gfalse(W(k)false) is ramified precisely at S∪Q; for q∈Q, ρq:GF→Gfalse(W(k)false) is unramified outside S∪Q∖{q} and ρ and ρq are congruent mod pd if ρ mod pd is unramified at q. Furthermore, the Galois representations {ρq} are ‘independent’.

Wiles Defect for Modules and Criteria for Freeness

“…Thus the A-module M is free. Theorem 4.6 implies the following known isomorphism criteria; see [9,Lemma A.8] and [7,Theorem 5.21]. We state it here for ease of reference as its needed later, and our methods yield a new proof of it.…”

“…One input in its proof is a result of [9] that dealt with the case M is a cyclic Amodule; see Theorem 4.8 below. The main new ingredient is the following criterion for freeness of modules.…”

Wiles defect of Hecke algebras via local-global arguments

“…Proof. We may apply Lemma A.10 of [FKR21] to the map φ C , since C is Cohen-Macaulay, to deduce that is an isomporphism since Ann A (ker(φ B )) ker(φ B ) = (0), so Ann A (ker(φ B )) ker(φ C ), being a submodule of a finite length A-module, is also of finite length. On the other hand, it is a submodule of A and depth(A) = 1, so it must be (0).…”