Characteristic p is rational flux: a Hofstadter reading of geometric Langlands

Characteristic p is rational flux: a Hofstadter reading of geometric Langlands in prime characteristic

Standalone note. Claim: the algebraic structure that makes geometric Langlands tractable in characteristic p (Braverman–Bezrukavnikov 2007, Chen–Zhu 2017; Bezrukavnikov–Mirković–Rumynin) is the same structure that makes the Hofstadter problem tractable at rational flux. Frobenius is the magnetic unit cell.

1. Three algebras with one shape

Rotation algebra at θ = p/q. Generators U, V with UV = e^{2πiθ} VU (magnetic translations by one lattice step, flux θ per plaquette). Then

U^q V = e^{2πi p} V U^q = V U^q,

so U^q and V^q are central: translations by the magnetic unit cell commute because the flux through it is the integer p (Dirac quantization for the sublattice). The algebra is a bundle of q×q matrix algebras over the torus Spec C*(U^q, V^q) — the magnetic Brillouin zone. The Hofstadter spectrum is the q bands of this bundle.

Weyl algebra in characteristic p. Generators x, ∂ with [∂, x] = 1. Then [∂, x^p] = p x^{p−1} = 0 and [∂^p, x] = 0, so x^p and ∂^p are central. The algebra is Azumaya of rank p² over its center k[x^p, ∂^p], i.e. a bundle of p×p matrix algebras over the Frobenius twist of the cotangent plane. Globally: crystalline differential operators D_X are Azumaya over T*X^{(1)} (BMR).

Quantum group at q^ℓ = 1. E^ℓ, F^ℓ, K^ℓ generate the big central subalgebra Z_0 (De Concini–Kac–Procesi); generic irreducibles have dimension ℓ^N.

Dictionary. Frobenius twist X^{(1)} ↔ magnetic Brillouin zone. Functions on X^{(1)} are p-th powers ↔ the center is generated by q-th powers. Azumaya ↔ matrix bundle. Generic ħ over ℂ ↔ irrational θ: the algebra is simple, there is no center, nothing descends.

2. p-curvature is the flux through the unit cell

For a connection ∇ on a curve in characteristic p, ψ(∂) = (∇_∂)^p − ∇_{∂^p} is 𝓞-linear: a Higgs field on X^{(1)}. For rank one, ∇ = d + f dt,

(∂ + f)^p = ∂^p + f^p + ∂^{p−1}(f) (Jacobson),

so ψ = f^p + ∂^{p−1}f. It measures the failure of the connection to descend along Frobenius — the only curvature that survives at the scale where translations commute. That is exactly the role of the flux through the enlarged cell in Hofstadter: the residual field seen by the commuting sublattice.

In Chen–Zhu, the Hitchin map on D-modules in characteristic p is the p-curvature, landing on the Frobenius-twisted Hitchin base, and the Langlands equivalence is a twisted Fourier–Mukai transform over that base using the Azumaya structure. In Hofstadter language: Fourier duality on the magnetic Brillouin zone, twisted by the matrix bundle.

3. Wild ramification: Swan = pole order of p-curvature − 1

Artin–Schreier y^p − y = t^{−m}, p ∤ m, ramification break m. The associated connection d + da has f = −m t^{−m−1}. The term ∂^{p−1}(t^{−m−1}) has coefficient (−m−1)(−m−2)···(−m−p+1), a product over every residue class mod p except −m, hence containing 0, hence zero. So

ψ = (−m)^p t^{−p(m+1)},

a pole of order m+1 on X^{(1)}. Swan conductor m = irregularity of the p-curvature. This is Deligne's Swan/irregularity comparison, here read as: wild ramification is irregular flux through the unit cell.

4. The Brauer invariant is flux per cell

A local division algebra of invariant k/n is the cyclic algebra (F_n/F, Frob, ϖ^k): a twisted group algebra of ℤ/n where the generator acts by Frobenius — the same shape as U, V. The rotation algebra A_{p/q} is Azumaya of degree q over T² with class p ∈ H²(T², μ_q) ≅ ℤ/q (the first Chern class of the underlying rank-q bundle, reduced mod q). Arithmetic: Br(F)[n] = H²(F, μ_n) ≅ ℤ/n is the invariant map. So the invariant is flux per cell mod n. The integer lift, the Hall conductance c₁ ∈ ℤ, is fixed in Hofstadter by the TKNN Diophantine equation and requires a base torus; the arithmetic home of such a lift is the Azumaya class of D_{Bun} over T*Bun^{(1)}.

5. Mixed characteristic is the neighborhood of rational flux

The Weyl algebra over ℚ_p has trivial center (characteristic 0), so p-adic fields are not at rational flux. The residue field is; the p-adic field is the deformation θ = p/q + ε, where Frobenius survives only as a semilinear structure (Dwork's F-isocrystals). Two echoes: Wilkinson's renormalization of the butterfly near rational flux (a q×q structure plus a small perturbation), and the p-adic local monodromy theorem (a Frobenius structure forces quasi-unipotent monodromy — a commuting sublattice reappears after a finite cover). On the Galois/Brauer side the picture is exactly rational (Br(F) is split by the maximal unramified extension); on the differential-equation side it is deformed.

6. Status

Sections 1–3 are theorems restated; the only claim is the identification of names across the two literatures, which I have not seen written down. Section 4 is a mod-n theorem plus an unverified integer-lift prediction. Section 5 is structural conjecture.