8 - Numerical polynomials and endomorphisms of formal group laws

Summary

Abstract. Endomorphism rings of formal group laws are used to construct families of polynomials which take integer values at the integers. Applications are made to the study of Hopf algebroids of stable cooperations for generalized homology theories and to the study of generalized Bernoulli numbers.

Primary 13B25, 14L05; Secondary 55N20, 11B68

A numerical polynomial is a polynomial with coefficients in ℚ which takes integer values at the integers, or more generally, with coefficients in a number field K taking values in A, the ring of integers of K, when evaluated at elements of A. The study of such polynomials has a long history and a substantial literature (for example). The connection with formal group laws and topology stems from the results of where the notion of a stably numerical Laurent polynomial was introduced: a Laurent polynomial f(x) ∈ ℚ[x, x⁻¹] is stably numerical if x^kf(x) is numerical for some integer k. The main result of is that the Hopf algebra of degree 0 stable co-operations for complex K-theory, K₀K, is isomorphic to the algebra of stably numerical Laurent polynomials for ℤ. This is essentially a result connecting stably numerical Laurent polynomials with the multiplicative formal group law, and this connection was described explicitly in. The point of the current paper is that this method is applicable to a large class of formal group laws, that its use provides a large supply of stably numerical Laurent polynomials, both for ℤ and for the integers in other number fields, and that when it is applied to the formal group laws associated to elliptic curves new and useful information about the Hopf algebroid of stable cooperations…