Augmentation-Ideal Realization of Special Nottingham Congruence Series and Cartier Indecomposables

Abstract

Let k = F_p with p >= 5, let A = k[[x_1,...,x_n]] with n >= 3, and let m be its maximal ideal. Let G be the group of volume-preserving formal automorphisms of A tangent to the identity, let G_s consist of those g in G such that g(x_i) - x_i lies in m^s for every i, and let T_N = G/G_N. The action of T_N on A/m^N defines, by operator order, a multiplicative filtration I_a of the integral augmentation ideal such that the dimension subgroup determined by I_a is exactly G_{a+1}/G_N. Thus the shifted congruence filtration of T_N is realized by two-sided ideals in Z[T_N]. The associated group symbols identify with the degree less than N truncation of the graded Lie algebra of divergence-free polynomial vector fields. If E_d denotes the subspace corresponding to exact (n-1)-forms, then [L_2,L_{d-1}] = E_d for 3 <= d < N. Hence the degree-d indecomposable quotient is L_d/E_d. This quotient vanishes unless d = (p-1)(n-1) + pr for some r >= 0. In those exceptional degrees it is naturally isomorphic, as a rational GL(V)-module, to the tensor product of (det V*)^(p-1) with the first Frobenius twist of Sym^r(V*) tensor V. These quotients are compatible with truncation in N.