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Home > Vol 17, No 1 (2003)

Jesús González

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) = _{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 _* = _{( 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 } ₁ in BP _* and studying the value of a _k,s in each of the associated (bi)graded rings BP _*/ W . This allows us to identify, among monomials in a _k,s of minimal W -filtration (1 k), an explicit monomial m _{, k,s} carrying the lowest possible p-divisibility. The p-local coefficient of m _{, 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 .

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