The Critical Cartier Diagonal in the Lower p-Central Series of Jacobian-One Formal Automorphisms

Abstract

Let k = F_p with p ≥ 5, let V = k^n with n ≥ 3, and let T(N) be a finite congruence quotient of the tangent-to-the-identity Jacobian-one formal automorphism group in n variables. Write P(1) = T(N), P(i + 1) = P(i)^p[P(i), T(N)], and d(r) = (p − 1)(n − 1) + pr. For every r ≥ 2 and N > d(r), put F(i,r) = gr_{d(r)} P(i). We determine the corresponding critical diagonal layers: F(r − 1,r) = L_{d(r)}, F(r,r) = Π(r)⁻¹(δ(1) ⊗ E(r)^(1)), where δ(j) = (det V*)^(p^j − 1), Π(r) is the rational GL(V)-equivariant Cartier map, W(r) = Sym^r(V*) ⊗ V, and E(r) is the coefficient-exact subspace of W(r). Consequently, gr_{d(r)}(P(r − 1)/P(r)) ≅ δ(1) ⊗ (W(r)/E(r))^(1) ≠ 0. The commutator subgroup already realises the entire denominator. Restricted powers occur nontrivially at the critical physical degree, but their symbols lie inside the exact physical subspace and therefore vanish in the critical quotient. We also determine the rational GL(V)-module structure of every critical layer. Away from Cartier resonance, the critical module is a Frobenius-twisted symmetric power. At the resonance indices r = (p − 1)(n − 1) + pq, it is governed by a natural four-term exact sequence involving the second Cartier descent.