{"id":"hardly_ramified_reducible","label":"hardly_ramified_reducible","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_3.html","dependencies":["hardly_ramified_lifts","hardly_ramified_spreads_out","hardly_ramified_3adic_reducible"],"nlp_text":"If \\(\\ell \\geq 3\\) is a prime and \\(\\rho :\\operatorname{Gal}(\\overline{\\mathbb {Q}}/\\mathbb {Q})\\to \\operatorname{GL}_2(\\mathbb {Z}/\\ell \\mathbb {Z})\\) is hardly ramified, then \\(\\rho \\) is reducible.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FreyCurve.torsion_not_isIrreducible","latex_link":"ch_freyreduction.html#hardly_ramified_reducible","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FreyCurve.torsion_not_isIrreducible","lean_decl_name":"FreyCurve.torsion_not_isIrreducible","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/GaloisRepresentation/HardlyRamified/Frey.lean","lean4_code":"theorem FreyCurve.torsion_not_isIrreducible :\n    haveI : Fact (P.p.Prime) := ⟨P.pp⟩\n    ¬ GaloisRep.IsIrreducible (P.freyCurve.galoisRep P.p P.hppos) :=\n  sorry -- TODO prove this","line":22,"end_line":25,"proof_status":"statement_formalized"}
{"id":"Wiles_Frey_again","label":"Wiles_Frey_again","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_3.html","dependencies":["hardly_ramified_reducible","Frey_curve_hardly_ramified"],"nlp_text":"If \\(\\overline{\\rho }\\) is the mod \\(p\\) Galois representation associated to a Frey package \\((a,b,c,p)\\) then \\(\\overline{\\rho }\\) is reducible.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Wiles_Frey","latex_link":"ch_freyreduction.html#Wiles_Frey_again","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Wiles_Frey","lean_decl_name":"Wiles_Frey","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch3","source_path":"FLT/Basic/Reductions.lean","lean4_code":"theorem Wiles_Frey (P : FreyPackage) :\n    haveI : Fact P.p.Prime := ⟨P.pp⟩\n    ¬ GaloisRep.IsIrreducible (P.freyCurve.galoisRep P.p P.hppos) :=\n  FreyCurve.torsion_not_isIrreducible P\n","line":47,"end_line":50,"proof_status":"statement_formalized"}
{"id":"hardly_ramified_lifts","label":"hardly_ramified_lifts","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_3.html","dependencies":["hardly_ramified"],"nlp_text":"If \\(\\ell \\geq 3\\) is prime and \\(\\overline{\\rho }:\\operatorname{Gal}(\\overline{\\mathbb {Q}}/\\mathbb {Q})\\to \\operatorname{GL}_2(\\mathbb {Z}/\\ell \\mathbb {Z})\\) is hardly ramified and irreducible, then there exists a finite extension \\(K\\) of \\(\\mathbb {Q}_\\ell \\) with integer ring \\(\\mathcal{O}\\) and maximal ideal \\(\\mathfrak {m}\\) and a hardly ramified representation \\(\\rho :\\operatorname{Gal}(\\overline{\\mathbb {Q}}/\\mathbb {Q})\\to \\operatorname{GL}_2(\\mathcal{O})\\) whose reduction modulo \\(\\mathfrak {m}\\) is isomorphic to \\(\\rho \\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/GaloisRepresentation.IsHardlyRamified.lifts","latex_link":"ch_freyreduction.html#hardly_ramified_lifts","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/GaloisRepresentation.IsHardlyRamified.lifts","lean_decl_name":"GaloisRepresentation.IsHardlyRamified.lifts","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch3","source_path":"FLT/GaloisRepresentation/HardlyRamified/Lift.lean","lean4_code":"/--\nAn irreducible mod p hardly ramified representation lifts to a p-adic one.\n-/\ntheorem lifts (ρ : GaloisRep ℚ k V) (hρirred : ρ.IsIrreducible)\n    (hρ : IsHardlyRamified hpodd hV ρ) :\n    ∃ (R : Type u) (_ : CommRing R) (_ : IsLocalRing R)\n      (_ : TopologicalSpace R) (_ : IsTopologicalRing R)\n      (_ : Algebra ℤ_[p] R) (_ : IsLocalHom (algebraMap ℤ_[p] R))\n      (_ : Module.Finite ℤ_[p] R) (_ : Module.Free ℤ_[p] R)\n      (_ : IsModuleTopology ℤ_[p] R)\n      (_ : Algebra R k) (_ : IsScalarTower ℤ_[p] R k) (_ : ContinuousSMul R k)\n      (W : Type v) (_ : AddCommGroup W) (_ : Module R W) (_ : Module.Finite R W)\n      (_ : Module.Free R W) (hW : Module.rank R W = 2)\n      (σ : GaloisRep ℚ R W) (r : k ⊗[R] W ≃ₗ[k] V),\n    IsHardlyRamified hpodd hW σ ∧ (σ.baseChange k).conj r = ρ := sorry\n","line":17,"end_line":31,"proof_status":"statement_formalized"}
{"id":"hardly_ramified","label":"hardly_ramified","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_3.html","dependencies":[],"nlp_text":"Let \\(R\\) be a coefficient ring with finite residue field of characteristic \\(\\ell \\geq 3\\). Let \\(V\\) be a finite free \\(R\\)-module of rank 2, equipped with the product topology. A continuous representation \\(\\rho : \\operatorname{Gal}(\\overline{\\mathbb {Q}}/\\mathbb {Q})\\to \\operatorname{GL}_R(V)\\) is said to be hardly ramified if it satisfies the following four conditions: \\(\\det (\\rho ):\\operatorname{Gal}(\\overline{\\mathbb {Q}}/\\mathbb {Q})\\to R^\\times \\) is the cyclotomic character; \\(\\rho \\) is unramified outside \\(2\\ell \\); The restriction of \\(\\rho \\) to \\(\\operatorname{Gal}(\\overline{\\mathbb {Q}}_2/\\mathbb {Q}_2)\\) is reducible (more precisely, there is a short exact sequence \\(0\\to R\\to V\\to R\\to 0\\) which is stable under the action of \\(\\operatorname{Gal}(\\overline{\\mathbb {Q}}_2/\\mathbb {Q}_2)\\)) and the Galois action on the 1-dimensional quotient is an unramified representation of \\(\\operatorname{Gal}(\\overline{\\mathbb {Q}}_2/\\mathbb {Q}_2)\\) whose square is trivial; The restriction of \\(\\rho \\) to \\(\\operatorname{Gal}(\\overline{\\mathbb {Q}}_\\ell /\\mathbb {Q}_\\ell )\\) is flat, by which we mean that for all open ideals \\(I\\) of \\(R\\), the (finite image) representation \\(\\rho \\) mod \\(I:\\operatorname{Gal}(\\overline{\\mathbb {Q}}_\\ell /\\mathbb {Q}_\\ell )\\to \\operatorname{GL}_{R/I}(V/I)\\) comes from a finite flat group scheme.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/GaloisRepresentation.IsHardlyRamified","latex_link":"ch_freyreduction.html#hardly_ramified","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/GaloisRepresentation.IsHardlyRamified","lean_decl_name":"GaloisRepresentation.IsHardlyRamified","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch3","source_path":"FLT/GaloisRepresentation/HardlyRamified/Defs.lean","lean4_code":"/-- Let `R` be a compact Hausdorff local topological ring (for example any complete Noetherian\nlocal ring with the maximal ideal-adic topology) having finite residue field of\ncharacteristic `ℓ > 2`, and let `ρ : Gal(Qbar/Q) → GL_2(R)` be a continuous 2-dimensional\nrepresentation. We say that `ρ` is *hardly ramified* if it has cyclotomic determinant, is\nunramified outside `2ℓ`, flat at `ℓ` and upper-triangular at 2 with a 1-dimensional quotient which\nis unramified and whose square is trivial. -/\nstructure IsHardlyRamified {ℓ : ℕ} [Fact ℓ.Prime] (hℓOdd : Odd ℓ)\n    {R : Type u} [CommRing R] [TopologicalSpace R] [IsTopologicalRing R] [IsLocalRing R]\n    [Algebra ℤ_[ℓ] R] --[IsLocalHom (algebraMap ℤ_[ℓ] R)] -- a convenient way of saying \"residue\n    -- field has char ell\"\n    -- Rather than GL_2(R) we use the automorphisms of a finite free rank 2 `R`-module `V`.\n    {V : Type*} [AddCommGroup V] [Module R V]\n    [Module.Finite R V] [Module.Free R V] (hdim : Module.rank R V = 2)\n  -- Let `ρ` be a continuous action of the absolute Galois group of `ℚ` on `V`.\n    (ρ : GaloisRep ℚ R V) : Prop where\n  -- We say `ρ` is *hardly ramified* if\n  -- `det(ρ)` is the ell-adic cyclotomic character;\n  det : ∀ g, ρ.det g = algebraMap ℤ_[ℓ] R (cyclotomicCharacter (ℚ ᵃˡᵍ) ℓ g.toRingEquiv)\n  -- `ρ` is unramified outside `2` and `ℓ`;\n  isUnramified : ∀ p (hp : p.Prime), p ≠ 2 ∧ p ≠ ℓ →\n    ρ.IsUnramifiedAt hp.toHeightOneSpectrumRingOfIntegersRat\n  -- ρ is flat at ℓ;\n  isFlat : ρ.IsFlatAt (Nat.Prime.toHeightOneSpectrumRingOfIntegersRat (Fact.out : ℓ.Prime))\n  -- and ρ has a 1-dimensional quotient π : ρ → δ such that\n  isTameAtTwo : ∃ (π : V →ₗ[R] R) (_ : Function.Surjective π) (δ : GaloisRep ℚ_[2] R R),\n    ∀ g : Γ ℚ_[2], ∀ v : V, π (ρ.map (algebraMap ℚ ℚ_[2]) g v) = δ g (π v) ∧\n    -- δ is unramified and\n    (AddSubgroup.inertia ((𝔪 Z2bar).toAddSubgroup : AddSubgroup Z2bar) (Γ ℚ_[2]) ≤ δ.ker) ∧\n    -- δ² = 1.\n    (∀ g : Γ ℚ_[2], δ g * δ g = 1)\n","line":80,"end_line":109,"proof_status":"formalized"}
{"id":"Frey_curve_hardly_ramified","label":"Frey_curve_hardly_ramified","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_3.html","dependencies":["hardly_ramified"],"nlp_text":"The \\(\\ell \\)-torsion \\(\\rho :\\operatorname{Gal}(\\overline{\\mathbb {Q}}/\\mathbb {Q})\\to \\operatorname{GL}_2(\\mathbb {Z}/\\ell \\mathbb {Z})\\) in the Frey curve associated to a Frey package \\((a,b,c,\\ell )\\) is hardly ramified.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FreyCurve.torsion_isHardlyRamified","latex_link":"ch_freyreduction.html#Frey_curve_hardly_ramified","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FreyCurve.torsion_isHardlyRamified","lean_decl_name":"FreyCurve.torsion_isHardlyRamified","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/GaloisRepresentation/HardlyRamified/Frey.lean","lean4_code":"theorem FreyCurve.torsion_isHardlyRamified :\n    haveI : Fact (P.p.Prime) := ⟨P.pp⟩\n    IsHardlyRamified P.hp_odd sorry\n      (P.freyCurve.galoisRep P.p (show 0 < P.p from P.hppos)) :=\n  sorry","line":16,"end_line":20,"proof_status":"statement_formalized"}
{"id":"hardly_ramified_spreads_out","label":"hardly_ramified_spreads_out","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_3.html","dependencies":["hardly_ramified"],"nlp_text":"If \\(\\ell \\geq 3\\) is prime, \\(K\\) is a finite extension of \\(\\mathbb {Q}_\\ell \\) with integers \\(\\mathcal{O}\\) and if \\(\\rho :\\operatorname{Gal}(\\overline{\\mathbb {Q}}/\\mathbb {Q})\\to \\operatorname{GL}_2(\\mathcal{O})\\) is a hardly ramified representation whose reduction is irreducible, then there exists a number field \\(M\\) and, for each finite place \\(\\mu \\) of \\(M\\) of characteristic prime to 2, with completion \\(M_\\mu \\) having integer ring \\(R_\\mu \\), a hardly ramified semisimple representation \\(\\rho _\\mu :\\operatorname{Gal}(\\overline{\\mathbb {Q}}/\\mathbb {Q})\\to \\operatorname{GL}_2(R_\\mu )\\) (by which we mean the generic fibre is semisimple), with the following properties: There is some \\(\\lambda \\mid \\ell \\) of \\(M\\) such that \\(\\rho _\\lambda \\cong \\rho \\), the isomorphism happening over some appropriate local field containing a copy of \\(M_\\lambda \\) and a copy of&#160;\\(K\\); If \\(\\mu _1\\) and \\(\\mu _2\\) are two finite places of \\(M\\) with odd residue characteristics \\(m_1\\) and \\(m_2\\), and if \\(p\\nmid 2m_1m_2\\) is prime, then \\(\\rho _{\\mu _1}\\) and \\(\\rho _{\\mu _2}\\) are both unramified at&#160;\\(p\\) and the characteristic polynomials \\(\\rho _{\\mu _1}(\\operatorname{Frob}_p)\\) and \\(\\rho _{\\mu _2}(\\operatorname{Frob}_p)\\) lie in \\(M[X]\\) and are equal.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/GaloisRepresentation.IsHardlyRamified.mem_isCompatible","latex_link":"ch_freyreduction.html#hardly_ramified_spreads_out","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/GaloisRepresentation.IsHardlyRamified.mem_isCompatible","lean_decl_name":"GaloisRepresentation.IsHardlyRamified.mem_isCompatible","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch3","source_path":"FLT/GaloisRepresentation/HardlyRamified/Family.lean","lean4_code":"theorem mem_isCompatible (hρ : IsHardlyRamified hpodd hv ρ) :\n    -- Then `ρ` lives in a compatible family of Galois representations\n    -- i.e., there's a family σ of 2-dimensional representations of Γ_ℚ\n    -- parametrised by maps from a number field M → ℚ_p-bar\n    ∃ (E : Type v) (_ : Field E) (_ : NumberField E) (σ : GaloisRepFamily ℚ E 2),\n    -- which are compatible, and\n    σ.isCompatible ∧\n    -- are \"hardly ramified\" for ℓ>2,\n    (∀ {ℓ : ℕ} (hℓ : Fact ℓ.Prime) (hℓodd : Odd ℓ) (φ : E →+* AlgebraicClosure ℚ_[ℓ]),\n      -- by which we mean that for a representation σ_φ in the family,\n      -- there's a hardly-ramified representation `τ` to GL_2(A)\n      -- for A a module-finite free ℤ_ℓ-algebra\n      ∃ (A : Type u) (_ : CommRing A) (_ : TopologicalSpace A) (_ : IsTopologicalRing A)\n        (_ : IsLocalRing A) (_ : Algebra ℤ_[ℓ] A) (_ : Module.Finite ℤ_[ℓ] A)\n        (_ : Module.Free ℤ_[ℓ] A) (_ : IsDomain A) (_ : Algebra A (AlgebraicClosure ℚ_[ℓ]))\n        (_ : IsScalarTower ℤ_[ℓ] A (AlgebraicClosure ℚ_[ℓ])) (_ : IsModuleTopology ℤ_[ℓ] A)\n        (_ : ContinuousSMul A (AlgebraicClosure ℚ_[ℓ]))\n        (W : Type v) (_ : AddCommGroup W) (_ : Module A W) (_ : Module.Finite A W)\n        (_ : Module.Free A W) (hW : Module.rank A W = 2)\n        (τ : GaloisRep ℚ A W)\n        (r : AlgebraicClosure ℚ_[ℓ] ⊗[A] W ≃ₗ[AlgebraicClosure ℚ_[ℓ]]\n          Fin 2 → AlgebraicClosure ℚ_[ℓ]),\n        IsHardlyRamified hℓodd hW τ ∧\n        -- whose base extension to GL_2(ℚ_p-bar) is φ_σ\n        (τ.baseChange (AlgebraicClosure ℚ_[ℓ])).conj r = σ hℓ φ) ∧\n    -- and `ρ` is part of the family.\n    (∃ (_ : Algebra R (AlgebraicClosure ℚ_[p])) (_ : ContinuousSMul R (AlgebraicClosure ℚ_[p]))\n      (ψ : E →+* AlgebraicClosure ℚ_[p])\n      (r' : AlgebraicClosure ℚ_[p] ⊗[R] V ≃ₗ[AlgebraicClosure ℚ_[p]]\n        Fin 2 → AlgebraicClosure ℚ_[p]),\n      (ρ.baseChange (AlgebraicClosure ℚ_[p])).conj r' = σ hp ψ) :=\n  sorry\n","line":21,"end_line":52,"proof_status":"statement_formalized"}
{"id":"hardly_ramified_mod3_reducible","label":"hardly_ramified_mod3_reducible","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_3.html","dependencies":["hardly_ramified"],"nlp_text":"Suppose \\(k\\) is a finite field of characteristic 3, and suppose \\(\\overline{rho}:\\operatorname{Gal}(\\overline{\\mathbb {Q}}/\\mathbb {Q})\\to \\operatorname{GL}_2(k)\\) is hardly ramified. Then \\(\\overline{\\rho }\\) is an extension of the cyclotomic character by the trivial representation.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/GaloisRepresentation.IsHardlyRamified.mod_three","latex_link":"ch_freyreduction.html#hardly_ramified_mod3_reducible","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/GaloisRepresentation.IsHardlyRamified.mod_three","lean_decl_name":"GaloisRepresentation.IsHardlyRamified.mod_three","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch3","source_path":"FLT/GaloisRepresentation/HardlyRamified/ModThree.lean","lean4_code":"/-- A mod 3 hardly ramified representation is an extension of trivial by cyclo -/\n-- Probably `Field k` can be replaced with `(3 : k) = 0`\ntheorem mod_three {k : Type u} [Fintype k] [Field k] [Algebra ℤ_[3] k] --\n    [TopologicalSpace k] [DiscreteTopology k]\n    (V : Type*) [AddCommGroup V] [Module k V] [Module.Finite k V] [Module.Free k V]\n    (hV : Module.rank k V = 2) {ρ : GaloisRep ℚ k V}\n    (hρ : IsHardlyRamified (show Odd 3 by decide) hV ρ) :\n    ∃ (π : V →ₗ[k] k) (_ : Function.Surjective π),\n    ∀ g : Γ ℚ, ∀ v : V, π (ρ g v) = π v := by\n  sorry\n","line":9,"end_line":18,"proof_status":"statement_formalized"}
{"id":"hardly_ramified_3adic_reducible","label":"hardly_ramified_3adic_reducible","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_3.html","dependencies":["hardly_ramified_mod3_reducible"],"nlp_text":"Suppose \\(L/\\mathbb {Q}_3\\) is a finite extension, with integer ring \\(\\mathcal{O}_L\\), and suppose \\(\\rho _3:\\operatorname{Gal}(\\overline{\\mathbb {Q}}/\\mathbb {Q})\\to \\operatorname{GL}_2(\\mathcal{O}_L)\\) is hardly ramified. Then (considered as a representation to \\(\\operatorname{GL}_2(L)\\)) \\(\\rho _3^{ss}=1\\oplus \\chi _3\\) where \\(1\\) is the trivial character and \\(\\chi _3\\) is the 3-adic cyclotomic character.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/GaloisRepresentation.IsHardlyRamified.three_adic","latex_link":"ch_freyreduction.html#hardly_ramified_3adic_reducible","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/GaloisRepresentation.IsHardlyRamified.three_adic","lean_decl_name":"GaloisRepresentation.IsHardlyRamified.three_adic","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch3","source_path":"FLT/GaloisRepresentation/HardlyRamified/Threeadic.lean","lean4_code":"/--\nA 3-adic hardly ramified representation has trace(Frob_p) = 1 + p for all p ≠ 2,3\n-/\ntheorem three_adic {R : Type*} [CommRing R] [Algebra ℤ_[3] R] [Module.Finite ℤ_[3] R]\n    [Module.Free ℤ_[3] R] [TopologicalSpace R] [IsTopologicalRing R] [IsLocalRing R]\n    [IsModuleTopology ℤ_[3] R]\n    (V : Type*) [AddCommGroup V] [Module R V] [Module.Finite R V] [Module.Free R V]\n    (hV : Module.rank R V = 2) {ρ : GaloisRep ℚ R V}\n    (hρ : IsHardlyRamified (show Odd 3 by decide) hV ρ) :\n    ∀ p (hp : Nat.Prime p) (hp5 : 5 ≤ p),\n      letI v := hp.toHeightOneSpectrumRingOfIntegersRat -- p as a finite place of ℚ\n      (ρ.toLocal v (Frob v)).trace _ _ = 1 + p := sorry\n","line":11,"end_line":22,"proof_status":"statement_formalized"}