For a positive integer
m
m
, set
ζ
m
=
exp
(
2
π
i
/
m
)
\zeta _{m}=\exp (2\pi i/m)
and let
Z
m
∗
\textbf {Z}_{m}^{*}
denote the group of reduced residues modulo
m
m
. Fix a congruence group
H
H
of conductor
m
m
and of order
f
f
. Choose integers
t
1
,
…
,
t
e
t_{1},\dots ,t_{e}
to represent the
e
=
ϕ
(
m
)
/
f
e=\phi (m)/f
cosets of
H
H
in
Z
m
∗
\textbf {Z}_{m}^{*}
. The Gauss periods
\[
θ
j
=
∑
x
∈
H
ζ
m
t
j
x
(
1
≤
j
≤
e
)
\displaylines { \theta _{j} =\sum _{x \in H} \zeta _{m}^{t_{j}x} \;\;\; (1 \leq j \leq e) }
\]
corresponding to
H
H
are conjugate and distinct over
Q
\textbf {Q}
with minimal polynomial
\[
g
(
x
)
=
x
e
+
c
1
x
e
−
1
+
⋯
+
c
e
−
1
x
+
c
e
.
\displaylines { g(x) = x^{e} + c_{1}x^{e-1} + \cdots + c_{e-1} x + c_{e}. }
\]
To determine the coefficients of the period polynomial
g
(
x
)
g(x)
(or equivalently, its reciprocal polynomial
G
(
X
)
=
X
e
g
(
X
−
1
)
)
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
m=p
, an odd prime, with
f
>
1
f >1
fixed. In this setting, it is known for certain integral power series
A
(
X
)
A(X)
and
B
(
X
)
B(X)
, that for any positive integer
N
N
\[
G
(
X
)
≡
A
(
X
)
⋅
B
(
X
)
p
−
1
f
(
mod
X
N
)
\displaylines { G(X) \equiv A(X)\cdot B(X)^{\frac {p-1}{f}} \;\;\;(\textrm {mod}\;X^{N}) }
\]
holds in
Z
[
X
]
\textbf {Z}[X]
for all primes
p
≡
1
(
mod
f
)
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
α
m=p^{\alpha }
, a prime power (
α
>
1
\alpha > 1
). The methods extend for odd prime powers
p
α
p^{\alpha }
to give a similar result for certain twisted Gauss periods of the form
\[
ψ
j
=
i
∗
p
∑
x
∈
H
(
t
j
x
p
)
ζ
p
α
t
j
x
(
1
≤
j
≤
e
)
,
\displaylines { \psi _{j} = i^{*} \sqrt {p} \sum _{x \in H} (\frac {t_{j}x}{p}) \zeta _{p^{\alpha }}^{t_{j}x} \;\;(1 \leq j \leq e),}
\]
where
(
p
)
(\frac { }{p})
denotes the usual Legendre symbol and
i
∗
=
i
(
p
−
1
)
2
4
i^{*}= i^{\frac {(p-1)^{2}}{4}}
.