On the minimal polynomial of Gauss periods for prime powers

Abstract:

For a positive integer $m$, set $\zeta _{m}=\exp (2\pi i/m)$ and let $\textbf {Z}_{m}^{*}$ 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 $e=\phi (m)/f$ cosets of $H$ in $\textbf {Z}_{m}^{*}$. The Gauss periods \[ \displaylines { \theta _{j} =\sum _{x \in H} \zeta _{m}^{t_{j}x} \;\;\; (1 \leq j \leq e) }\] corresponding to $H$ are conjugate and distinct over $\textbf {Q}$ with minimal polynomial \[ \displaylines { g(x) = x^{e} + c_{1}x^{e-1} + \cdots + c_{e-1} x + c_{e}. }\] To determine the coefficients of the period polynomial $g(x)$ (or equivalently, its reciprocal polynomial $G(X)=X^{e}g(X^{-1}))$ is a classical problem dating back to Gauss. Previous work of the author, and Gupta and Zagier, primarily treated the case $m=p$, 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 $N$ \[ \displaylines { G(X) \equiv A(X)\cdot B(X)^{\frac {p-1}{f}} \;\;\;(\textrm {mod}\;X^{N}) }\] holds in $\textbf {Z}[X]$ for all primes $p \equiv 1(\textrm {mod}\; f)$ except those in an effectively determinable finite set. Here we describe an analogous result for the case $m=p^{\alpha }$, a prime power ($\alpha > 1$). The methods extend for odd prime powers $p^{\alpha }$ to give a similar result for certain twisted Gauss periods of the form \[ \displaylines { \psi _{j} = i^{*} \sqrt {p} \sum _{x \in H} (\frac {t_{j}x}{p}) \zeta _{p^{\alpha }}^{t_{j}x} \;\;(1 \leq j \leq e),} \] where $(\frac { }{p})$ denotes the usual Legendre symbol and $i^{*}= i^{\frac {(p-1)^{2}}{4}}$.