Modified Stirling Numbers and p -Divisibility in the Universal Typical p k -Series

Abstract

The 1-dimensional universal formal group law is a power series (in two variables and with coefficients in Lazard's ring) carrying a lot of geometrical and algebraic properties. For a prime p, we study the corresponding “p-localized” formal group law through its associated p k-series, [p k](x) = Σ s≥0 a k,s x s(p−1)+1—the p k-fold iterated formal sum of a variable x. The coefficients a k,s lie in the Brown-Peterson ring BP * = ℝ(p)[v 1,v 2,...] and we describe part of their structure as polynomials in the variables v i with p-local coefficients. This is achieved by introducing a family of filtrations {W ϕ}ϕ≥1 in BP * and studying the value of a k,s in each of the associated (bi)graded rings BP */W ϕ. This allows us to identify, among monomials in a k,s of minimal W ϕ-filtration (1 ≤ ϕ ≤ k), an explicit monomial m ϕ,k,s carrying the lowest possible p-divisibility. The p-local coefficient of m ϕ,k,s is described as a Stirling-type number of the second kind and its actual value is computed up to p-local units. It turns out that m k,k,s not only carries the lowest W k -filtration but, more importantly, the lowest p-divisibility among all other monomials in a k,s . In particular, we obtain a complete description of the p-divisibility properties of each a k,s .