p_adic_langlands_conjecture

⚠️ Retired — incorrect formalization

The Lean statement below does not express the result p_adic_langlands_conjecture is named for, so its Proved status carries no information about it. Do not import it or use it as a dependency.

p-adic Langlands program: For GL₂(ℚₚ), there is a correspondence between p-adic representations of Gal(ℚ̄ₚ/ℚₚ) and p-adic Banach space representations of GL₂(ℚₚ). Proved for GL₂ (Breuil-Mézard, Colmez, Kisin); general GLₙ and other groups open.

Why this node was retired

The posted statement is

import Mathlib

theorem p_adic_langlands_conjecture (p : ℕ) (hp : Nat.Prime p) (n : ℕ) (hn : 1 ≤ n) :
    ∀ (rho : (ZMod p → ZMod p) → Matrix (Fin n) (Fin n) (ZMod p)),
      ∃ (pi : Matrix (Fin n) (Fin n) (ZMod p) → ℤ),
        True := by
  sorry

The goal is ∀ rho, ∃ (pi : Matrix (Fin n) (Fin n) (ZMod p) → ℤ), True, witnessed by the constant-zero function. No Galois representation, no p-adic Banach representation and no correspondence between them appears; rho and pi are arbitrary maps between finite matrix types.

What a faithful statement would require

Genuine Galois representations and p-adic representations of GL₂(ℚ_p) are needed, together with the assertion that the correspondence between them is a bijection with the expected compatibilities.

No corrected replacement node exists yet.