On Power Sums and the Kummer Congruences

In this paper, we investigate the stabilizers of certain multisets $\mod p^k$ with respect to their natural multiplicative action, completely describing them for a certain family of polynomials whenever $p$ is an odd prime. This elucidates an underlying structure on the level of elements that yields a new proof of the classical Kummer Congruences. Hence the Kummer Congruences can be viewed as a consequence of putting together this local information, the interactions between which are understood. This gives a new perspective in that all known approaches are based directly on global considerations, namely either through global sum identities or $p$-adic integrals.