The First Frattini–Commutator Layer of Jacobian-One Formal Automorphisms

Abstract

Let k = F_p with p >= 5, let V = k^n with n >= 3, and let G_1 be the tangent-to-the-identity Jacobian-one formal automorphism group, with congruence quotients T_N and Frattini subgroups Phi_N. We determine the first Frattini-commutator layer J_N = [T_N,Phi_N] / ([T_N,[T_N,Phi_N]] [T_N,Phi_N]^p). It is canonically the cokernel of H_2(T_N,k) -> H_2(T_N/[T_N,Phi_N],k). For d_r = (p - 1)(n - 1) + pr and N > d_4, its congruence-associated graded is gr_4 J_N isomorphic to L_4, gr_{d_2} J_N isomorphic to L_2^(1), gr_{d_4} J_N isomorphic to Sym^3(V*)^(1), with all other graded pieces vanishing. Hence dim_k J_N = n binomial(n + 3,4) + n(n - 1)(n + 2)/2, independently of N, and the truncation maps are eventually isomorphisms. Moreover, J_N isomorphic to P_3/P_4, where P_1 = T_N and P_{i+1} = P_i^p[P_i,T_N]. The vanishing above index four follows from zero-flux quadratic lifts and the surjectivity [L_2,W_{r-1}] = W_r. We also determine the first Cartier components of the associated commutator symbols; all of them vanish in J_N.