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f205175 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 | import Mathlib
import OAI.Probability.CriticalPercolation.LinearGroups.FgMatrixRing
namespace OAI
section
noncomputable section
open Set Function Module
open scoped Classical
namespace CriticalPercolation.Arithmetic
lemma exists_finite_padic_matrix_model {G n : Type} [Group G] [Group.FG G]
[Fintype n] [DecidableEq n] (p : ℕ) [Fact p.Prime]
(ρ : G →* GL n (AlgebraicClosure (Padic p)))
(E : Finset (AlgebraicClosure (Padic p))) :
∃ L : IntermediateField (Padic p) (AlgebraicClosure (Padic p)),
FiniteDimensional (Padic p) L ∧ (E:Set (AlgebraicClosure (Padic p)))⊆(L:Set (AlgebraicClosure (Padic p))) ∧
∃ τ : G →* GL n L, ∀ g,
Matrix.GeneralLinearGroup.map L.val.toRingHom (τ g)=ρ g := by
obtain ⟨D,hD,hm⟩ := LinearGrowth.exists_finiteType_matrix_ring ρ
obtain ⟨F,hF⟩ := (Subalgebra.fg_iff_finiteType D).mpr hD
let L := IntermediateField.adjoin (Padic p) (↑(F∪E):Set (AlgebraicClosure (Padic p)))
have hfin : FiniteDimensional (Padic p) L :=
IntermediateField.finiteDimensional_adjoin (fun x _ => Algebra.IsIntegral.isIntegral x)
have hDL : D≤L.toSubalgebra.restrictScalars ℤ := by
rw [←hF]
apply Algebra.adjoin_le
intro x hx
exact IntermediateField.subset_adjoin (Padic p) _ (Finset.mem_union_left _ hx)
let τ : G →* GL n L := LinearGrowth.liftMatrixGroup ρ
(L.toSubalgebra.restrictScalars ℤ) (fun g i j => hDL (hm g i j))
refine ⟨L,hfin,?_,τ,?_⟩
· intro x hx
exact IntermediateField.subset_adjoin (Padic p) _ (Finset.mem_union_right _ hx)
· intro g
exact LinearGrowth.liftMatrixGroup_map ρ (L.toSubalgebra.restrictScalars ℤ) _ g
end CriticalPercolation.Arithmetic
end
end
end OAI
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