are 
, 
, and
. We do this by counting the G-orbits in the vector space V over
the finite field 
,
of order 
, for every 
. We use estimates of the number
of such points for orbits of a specific dimension to show there are
no further orbits of dimension 14 or higher.
We recall familiar general facts (usually known as Galois cohomology),
concerning forms which are
inequivalent over 
but
become
equivalent when viewed as forms over k.
We are assuming here that 
where k is the algebraic
closure of 
, and that 
for some a.
Let 
be the Frobenius map which raises elements to the q-th power, and
let it act entrywise on G. The set of fixed points under F in G
is 
of order
Let W be the
16-dimensional vector space of forms modulo cubics we have been
analyzing, but now defined over 
, so that 
.
We want to consider an orbit Gf say of the form f. We
assume here that f has coefficients fixed by F; in other words,
they lie in 
, which is the fixed field of F.
For f as in Table 1, they are in the ground field 
. We
will try to analyze the set Y of elements in Gf which are fixed by
F, that is, 
.
We are using the two actions of F on G and also on V.
Let 
. If gf
has coefficients fixed by F, so does hgf for each 
.
This means that the elements of H
act on Y and form orbits under this action. As 
,
the cardinality of Y is finite.
Let R be the stabilizer in G of f and S its connected component.
Then 
is finite, say of size n, so there
are coset representatives 
of S in R.
We assume that q has been
chosen so they are fixed by F. Assume 
is a complete set of
representatives for the conjugacy classes of 
. We prove the following:
Theorem. There are exactly r orbits of H acting on Y.
The orbits are precisely
, 
, where 
with
. Moreover, if
denotes the subgroup of R consisting of
all elements 
for which 
, the number
of elements in 
is 
.
We proceed in five steps.
A.
.
Notice first that 
,
as f has coefficients in 
. It follows that
if 
, then
,
and so 
so 
for
and 
. Suppose on the other hand that
for some i, 
and s in S;
then 
. This means that all elements in Y are of the
form gf for such a g.
B. For each i, there is
a 
for which 
.
Lang's theorem applies to
G acting on itself as G is connected. The fixed points of
the automorphism F are finite and so there is a 
for which
.
C. 
.
If 
, then 
.
To see the converse suppose 
satisfies
. Then
and so 
.
This means the coset 
is exactly the set of elements g of
G for which 
. There is one orbit of H in
Y given by 
.
D. Suppose that 
satisfies
with 
and 
for some
i. Then 
.
We will show that there exists
satisfying 
. Then
by C.
Notice that
,
and so we want to find 
with
or, equivalently,
Let 
be conjugation by t acting on S by
. The condition for 
is
Let 
. Note that
since 
, the Frobenius map F stabilizes R and so S as well.
By Lang's theorem,
all elements in the coset FG are conjugate in 
.
Thus
fixes only finitely many
points (indeed |H| points) on G and so also on S.
By Lang's theorem, applied to 
acting on S, there is such an 
.
E.
For 
, we have
if and only if 
and 
are conjugate in 
.
Suppose 
and 
are conjugate in 
by an element
. This means 
for 
.
Then 
. From D.,
it follows that 
. Note that 
and so the
H-orbits of 
and of 
are the same.
Suppose on the other hand that 
and 
are in the same
H-orbit. We may assume 
for some 
and
so 
for one of the coset representatives t and
. Now 
and so
. But under F, the coset tS is mapped
to tS as S is mapped to S and so
mod S, 
and 
are conjugate.
This means we need only consider the orbits 
.
F. The only remaining part is to show that the number of
elements in the orbit of 
is as indicated. The stabilizer
in H of 
is exactly 
. But,
for 
, the conjugate 
is in H if and only if it is fixed by F. This gives
which is
equivalent to 
as given, that is, 
.
Corollary. Suppose the stabilizer R splits
as an extension over S (so that we can choose
the coset representatives
of S in R in such a way that they form a
group themselves fixed by F).
Then
As above, let 
be the set of elements r in R
satisfying 
, so that 
.
Suppose 
. Write
with 
. Then 
Because
we have chosen the 
to be fixed by F,
this means 
. But this
implies that 
and 
commute modulo S and so commute
as the 
form a group.
This implies 
.
Thus, 
where
is the centralizer of 
in 
.
By the theorem, we now have
.
But 
is the size of the conjugacy class and so the result
follows.
Corollary. If R is finite, then
|Y|=|H|.
|S|=1 in the above.
Corollary. If 
(in particular
if R=S), then
(where 
are the just the fixed
points of F on S).
Any 
induces an inner
automorphism on S. It follows by Lang's theorem,
that 
is conjugate to F by an element of S and so has the
same number of fixed points as F on S.
We now add the contributions to 
from the orbits of forms 
, 
.
Note that the stabilizers in G of these forms are given in
.2. Here h=|H| is as in the beginning of this section.
For 
,
we shall denote by 
the set 
.
The stabilizer of 
is finite and so, by Corollary 6.3,
the number of points in 
is h.
The stabilizer of 
is 
where T is a
one-dimensional
torus and 
is the cyclic group of order 2
generated by the matrix 
. There are two
orbits, each of size 
, so 
.
The stabilizer of 
is connected and so the number of orbits is
one and the size is 
.
Now adding the contributions for these three orbits and subtracting
from 
gives a polynomial of degree 13.
We now quote a result by [LW] which shows there are no more orbits of dimension 14 or higher.
Theorem. [LW]. Let X be a d-dimensional
subvariety of an n-dimensional projective space defined
over 
. There exists a constant c (depending only on
X, d and n - in fact, not on X explicitly, but
on the degree of X in the projective space)
with 
.
Note this result also works
for subvarieties of affine space.
For one direction, embed the affine variety in
projective space and consider the closure;
then 
where 
is the number
of 
-points of the closure and use the upper bound
for that. For the other bound, choose an open subset O
contained in the original variety. Then the number of 
-points
on the closure 
of O
and the complement of O in 
can be estimated, and so also
on O; this gives the appropriate lower bound for 
.
Corollary.
Theorem 
holds.
If 
satisfies 
, then, by Theorem 6.5, the number of 
-points of Gf
(the only dense orbit of 
) is at least
(for some constant c), which contradicts
the fact that the number of 
-points of 
is a polynomial in q of degree 13. This shows
there are no further orbits of dimension 14 or higher. In
particular, all other orbits lie in 
. In , any form in
was shown to be listed in Table 1.
Our work is over the algebraic closure of 
. However arguments
in [GLMS] show the same applies over any algebraically closed field of
characteristic 3.
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