ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

3. Number Theory and Arithmetic Algebraic Geometry

1991 MS Classification: 11T22, 11T24 Gurak, Stanley, University of San Diego, USA On the factors Φ^{(jδ/m)}(x) of the period polynomial for finite fields. Denote by F_{q} the finite field of q elements, where q = ef + 1 is a power of a prime p. Set δ = gcd((q − 1)/(p − 1), e). The period polynomial Φ(x) of degree e for the finite field F_{q} is known to factor over the rational field Q as a product of δ polynomials, Φ^{(w)}(x) (1 ≤ w ≤ δ), each of degree e/δ. For f fixed, I obtained formulas for the beginning coefficients of Φ^{(δ)}(x) expressed in terms of a certain counting function β_{K}(n) for positive integers n, where K is the decomposition field of p in the cyclotomic field of f-roots of unity. These formulas are computationally impractical in general, but I later found closed form expressions or generating functions for β_{K}(n) when K = Q, which lead to explicit expressions for the beginning coefficients of Φ^{(δ)}(x). Subsequently, I found similar results for the middle factor Φ^{(δ/2)}(x), when δ is even. Here I examine the factors Φ^{(w)}(x), where w = jδ/m, m|δ and gcd(j, m) = 1, to obtain analogous results, particularly in the case K = Q.