Modified Stirling numbers and $p$-divisibility in the universal typical $p^k$-series

González, Jesús

Journal of Algebraic Combinatorics, Tome 17 (2003) no. 1, pp. 75-89.

Résumé

Summary: 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 $ldquop$-localized $rdquo$ formal group law through its associated $p ^{ k }$-series, $[ p ^{k}]( x) = Sgr_{ $sge$0}$ 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 _*$ = Ropf_{( p)}[ v _{1}, v _{2},\dots ]$ 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 $W \_$ varphi _ phivge$1}$ in BP _* and studying the value of a $\_{ k,s }$ in each of the associated (bi)graded rings BP $\_{*}/ W \_$ varphi. This allows us to identify, among monomials in a $\_{ k,s }$ of minimal W _ varphi-filtration (1 $lephivlek$), an explicit monomial $m \_$ varphi, k,s carrying the lowest possible p-divisibility. The p-local coefficient of m _ varphi, 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 $\_{ k,s }$.

Détail

Keywords: formal group laws, universal typical $p ^{ k }$-series, Stirling numbers