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Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

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3 declarations, 0 contain sorry, 0 declared as axioms

Hardly ramified Galois representations

We define what it means for a continuous mod-ℓ (or ℓ-adic) representation of Gal(ℚ̄/ℚ) to be hardly ramified: unramified outside 2ℓ, with prescribed flat / finite ramification at 2 and ℓ. This is the precise hypothesis satisfied by the ℓ-torsion of the Frey curve.

Imports (5)

Imported by 6 files

abbrev GaloisRepresentation.Z2bar Z2bar is the ring of integers of ℚ_[2]ᵃˡᵍ line 78 source
Source documentation
Z2bar is the ring of integers of ℚ_[2]ᵃˡᵍ.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: ValuationSubring, AlgebraicClosure, Padic, OfNat.ofNat, Nat
Proof uses: Valuation.valuationSubring, NNReal, Valued.v
12 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
```
noncomputable abbrev Z2bar : ValuationSubring (ℚ_[2]ᵃˡᵍ) := Valued.v.valuationSubring
```
instance GaloisRepresentation.(instance at line 80) line 80 source
structure GaloisRepresentation.IsHardlyRamified Documented declaration: IsHardlyRamified line 96 source
Source documentation
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.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, Odd, CommRing, TopologicalSpace, IsTopologicalRing, IsLocalRing, Algebra, PadicInt, AddCommGroup, Module, Module.Finite, Module.Free, Eq, Cardinal, Module.rank, OfNat.ofNat, GaloisRep, Rat
14 instances, coercions and auxiliary constants left out.
```
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
…
```
Mentions: GaloisRep