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.
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.
![[Annotate]](/organics/icons/sannotate.gif){kind=icon}
![[Shownotes]](../gif/annotate/shide-101.gif)