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

Abstract

The shifted congruence series of finite special Nottingham jet groups is realized by an integral augmentation filtration. For volume-preserving formal substitutions in characteristic $p\ge5$ and $n\ge3$ variables, the filtration induced by the action on truncated power series has the congruence subgroups as its dimension subgroups. The canonical weighted augmentation filtration has the same dimension subgroups. The group symbols form a degree-truncated Lie algebra of divergence-free polynomial fields. Since brackets with quadratic fields give the subspace of exact fields in every coefficient degree at least three, the Lie indecomposables occur only in degrees $(p-1)(n-1)+pr$, with $r\ge0$, where they are determinant-weighted Frobenius twists of tensor products of symmetric powers with the defining module.