Submit proof — GaloisRepresentation.IsHardlyRamified

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Posted 13 days ago

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/-- Let `R` be a compact Hausdorff local topological ring (for example any complete Noetherian
local ring with the maximal ideal-adic topology) having finite residue field of
characteristic `ℓ > 2`, and let `ρ : Gal(Qbar/Q) → GL_2(R)` be a continuous 2-dimensional
representation. We say that `ρ` is *hardly ramified* if it has cyclotomic determinant, is
unramified outside `2ℓ`, flat at `ℓ` and upper-triangular at 2 with a 1-dimensional quotient which
is unramified and whose square is trivial. -/
structure IsHardlyRamified {ℓ : ℕ} [Fact ℓ.Prime] (hℓOdd : Odd ℓ)
    {R : Type u} [CommRing R] [TopologicalSpace R] [IsTopologicalRing R] [IsLocalRing R]
    [Algebra ℤ_[ℓ] R] --[IsLocalHom (algebraMap ℤ_[ℓ] R)] -- a convenient way of saying "residue
    -- field has char ell"
    -- Rather than GL_2(R) we use the automorphisms of a finite free rank 2 `R`-module `V`.
    {V : Type*} [AddCommGroup V] [Module R V]
    [Module.Finite R V] [Module.Free R V] (hdim : Module.rank R V = 2)
  -- Let `ρ` be a continuous action of the absolute Galois group of `ℚ` on `V`.
    (ρ : GaloisRep ℚ R V) : Prop where
  -- We say `ρ` is *hardly ramified* if
  -- `det(ρ)` is the ell-adic cyclotomic character;
  det : ∀ g, ρ.det g = algebraMap ℤ_[ℓ] R (cyclotomicCharacter (ℚ ᵃˡᵍ) ℓ g.toRingEquiv)
  -- `ρ` is unramified outside `2` and `ℓ`;
  isUnramified : ∀ p (hp : p.Prime), p ≠ 2 ∧ p ≠ ℓ →
    ρ.IsUnramifiedAt hp.toHeightOneSpectrumRingOfIntegersRat
  -- ρ is flat at ℓ;
  isFlat : ρ.IsFlatAt (Nat.Prime.toHeightOneSpectrumRingOfIntegersRat (Fact.out : ℓ.Prime))
  -- and ρ has a 1-dimensional quotient π : ρ → δ such that
  isTameAtTwo : ∃ (π : V →ₗ[R] R) (_ : Function.Surjective π) (δ : GaloisRep ℚ_[2] R R),
    ∀ g : Γ ℚ_[2], ∀ v : V, π (ρ.map (algebraMap ℚ ℚ_[2]) g v) = δ g (π v) ∧
    -- δ is unramified and
    (AddSubgroup.inertia ((𝔪 Z2bar).toAddSubgroup : AddSubgroup Z2bar) (Γ ℚ_[2]) ≤ δ.ker) ∧
    -- δ² = 1.
    (∀ g : Γ ℚ_[2], δ g * δ g = 1)

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