Modified Stirling numbers and \(p\)-divisibility in the universal typical \(p^k\)-series (Q1868383)

This paper studies the coefficients of the multiplication by \(p^k\) map, \([p^k](x) = \sum a_{k,s} x^{s(p-1)+1}\) for the \(1\)-dimensional universal \(p\)-typical formal group law. By a well-known result of Quillen this is the formal group law associated to Brown Peterson cohomology, \(BP^*(\;)\), and so results about these coefficients have applications in algebraic topology. The coefficients lie in \(BP_*\) and so can be expressed in terms of the Araki generators as \(a_{k,s} = \sum c_{k,s,I} v^I\). The author previously [J. Pure Appl. Algebra 157, 57--68 (2001; Zbl 0966.55005)] computed a lower bound for the \(p\)-adic valuation of the coefficients \(c_{k,s,I}\) and \(a_{k,s}\). In the present paper he computes some of the \(c_{k,s,I}\)'s exactly up to \(p\)-local units and shows that his bound on the \(a_{k,s}\)'s is sharp. These results are proved by means of a new filtration on \(BP_*\) which is a modification of the usual one by powers of the invariant prime ideal.