On the minimal polynomial of Gauss periods for prime powers
Abstract
For a positive integer m , set ζ_{m}Dexp(2π i/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 t_{1},dots,t_{e} to represent the eDφ(m)/f cosets of H in {Z}_{m}^{*} . The Gauss periods θ_{j} Dsum_{x in H} ζ_{m}^{t_{j}x} ; (1 ≤ j ≤ e) corresponding to H are conjugate and distinct over {Q} with minimal polynomial g(x) = x^{e} + c_{1}x^{e1} + \cdots + c_{e1} x + c_{e}. To determine the coefficients of the period polynomial g(x) (or equivalently, its reciprocal polynomial G(X)DX^{e}g(X^{1})) is a classical problem dating back to Gauss. Previous work of the author, and Gupta and Zagier, primarily treated the case mDp , 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)\cdot B(X)^{{p1}/{f}} ({mod};X^{N}) 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 mDp^{α} , a prime power ( α > 1 ). The methods extend for odd prime powers p^{α} to give a similar result for certain twisted Gauss periods of the form ψ_{j} = i^{*} √{p} sum_{x in H} ({t_{j}x}/{p}) ζ_{p^{α}}^{t_{j}x} (1 ≤ j ≤ e), where ({ }/{p}) denotes the usual Legendre symbol and i^{*}D i^{{(p1)^{2}}/{4}} .
 Publication:

Mathematics of Computation
 Pub Date:
 December 2006
 DOI:
 10.1090/S0025571806018850
 Bibcode:
 2006MaCom..75.2021G