In this paper we prove that certain products and sums of powers of binomial coefficients modulo
p
=
q
f
+
1
p = qf + 1
,
q
=
a
2
+
b
2
q = {a^2} + {b^2}
, are determined by the parameters x occurring in distinct solutions of the quaternary quadratic partition
\[
16
p
α
=
x
2
+
2
q
u
2
+
2
q
v
2
+
q
w
2
,
(
x
,
u
,
v
,
w
,
p
)
=
1
,
x
w
=
a
v
2
−
2
b
u
v
−
a
u
2
,
x
≡
4
(
mod
q
)
,
α
⩾
1.
\begin {array}{*{20}{c}} {16{p^\alpha } = {x^2} + 2q{u^2} + 2q{v^2} + q{w^2},\quad (x,u,v,w,p) = 1,} \\ {xw = a{v^2} - 2buv - a{u^2},\quad x \equiv 4\pmod q,\alpha \geqslant 1.} \\ \end {array}
\]
The number of distinct solutions of this partition depends heavily on the class number of the imaginary cyclic quartic field
\[
K
=
Q
(
i
2
q
+
2
a
q
)
,
K = Q\left ( {i\sqrt {2q + 2a\sqrt q } } \right ),
\]
as well as on the number of roots of unity in K and on the way that p splits into prime ideals in the ring of integers of the field
Q
(
e
2
π
i
p
/
q
)
Q({e^{2\pi ip/q}})
. Let the four cosets of the subgroup A of quartic residues be given by
c
j
=
2
j
A
,
j
=
0
,
1
,
2
,
3
{c_j} = {2^j}A,j = 0,1,2,3
, and let
\[
s
j
=
1
q
∑
t
∈
c
j
t
,
j
=
0
,
1
,
2
,
3.
{s_j} = \frac {1}{q}\sum \limits _{t \in {c_j}} {t,\quad j = 0,1,2,3.}
\]
Let
s
m
{s_m}
and
s
n
{s_n}
denote the smallest and next smallest of the
s
j
{s_j}
respectively. We give new, and unexpectedly simple determinations of
Π
k
∈
c
n
k
f
!
{\Pi _{k \in {c_n}}}kf!
and
Π
k
∈
c
n
+
2
k
f
!
{\Pi _{k \in {c_{n + 2}}}}kf!
, in terms of the parameters x in the above partition of
16
p
α
16{p^\alpha }
, in the complicated case that arises when the class number of K is
>
1
> 1
and
s
m
≠
s
n
{s_m} \ne {s_n}
.