On the minimal polynomial of Gauss periods for prime powers

For a positive integer m, set ζm = exp(2πi/m) and let Z ∗ m denote the group of reduced residues modulo m. Fix a congruence group H of conductor m and of order f . Choose integers t1, . . . , te to represent the e = φ(m)/f cosets of H in Zm. The Gauss periods θj = ∑ x∈H ζ tjx m (1 ≤ j ≤ e) corresponding toH are conjugate and distinct over Q with minimal polynomial g(x) = x + c1x e−1 + · · ·+ ce−1x+ ce. To determine the coefficients of the period polynomial g(x) (or equivalently, its reciprocal polynomial G(X) = Xeg(X−1)) is a classical problem dating back to Gauss. Previous work of the author, and Gupta and Zagier, primarily treated the case m = p, an odd prime, with f > 1 fixed. In this setting, it is known for certain integral power series A(X) and B(X), that for any positive integer N G(X) ≡ A(X) ·B(X) p−1 f (mod X ) holds in Z[X] for all primes p ≡ 1(mod f) except those in an effectively determinable finite set. Here we describe an analogous result for the case m = pα, a prime power (α > 1). The methods extend for odd prime powers pα to give a similar result for certain twisted Gauss periods of the form ψj = i ∗√p ∑ x∈H ( tjx p )ζ tjx pα (1 ≤ j ≤ e), where ( p ) denotes the usual Legendre symbol and i∗ = i (p−1)2 4 .