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The remaining orbits and the finite case


In this section we show that the only remaining orbits, i.e., those in 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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