THE RESULTS OF THE PAPER:
You might be most
interested in the problem related to coding theory, that is
describing explicitly those curves that have the property of exactly qt+1
points over infinitely many of the finite fields Fqt.
This property is complementary to that of curves that
achieve the extremal bound for rational points over finite
fields. Often the same curve works for both
problems – although for different values of t.
We use the classification of finite simple groups and covering theory
in positive characteristic to do much more than solve
Carlitz's conjecture (1966). An exceptional polynomial f over a finite
field Fq is a
polynomial that is a permutation polynomial on
infinitely many finite extensions of Fq.
Carlitz's conjecture says f
must be have odd degree (if q is odd). It is immediate that you can
reduce to considering the case that f is indecomposable over Fq.
What we do is figure out the
nature of the monodromy groups of exceptional polynomials in positive
characteristic. The method for the original Schur conjecture shows
quickly that the ramification over the infinite prime is wild (the
prime characteristic divides the order of the inertia group). In
particular, much more than their just having odd degrees, except in the
case the characteristic is 2 or 3, those degrees actually are powers of
the
characteristic.
The more precise results are as follows. Recall an affine group – in
what appears below – is a semidirect product of a vector space V over Fp and a
subgroup of the general linear group GL(V) acting on V. In our results, pu
= q, for
some integer u.
- Excluding characteristic 2 and 3, arithmetic monodromy
groups of
exceptional polynomials must be affine groups with the degree the order
of V.
- We completely
classify exceptional polynomials of degree equal to the
characteristic, an unsolved problem from Dickson's thesis
(1896).
- Further, by generalizing Dickson's problem we describe
all known exceptional polynomials. Especially this includes Cohen's
semi-linear
polynomials. They all fit a simple criterion in our use of monodromy
groups.
- In the case of characteristic 2 and 3 we show precisely
what those arithmetic monodromy groups must be, but we didn't try to
find out if there were polynomials that realized those particular
groups.
TWO KEY IDEAS IN THE
PAPER:
Total, but wild, ramification: There is a key connection between geometry of covers and group theory.
As in characteristic 0 when dealing with polynomial covers, f, the total
ramification over ∞ is the key to much computation. There, and in characteristics prime
to the degree, the inertia group I∞ is cyclic of order the degree.
In all cases the investigation is to draw conclusions on the arithmetic Gf^ (and geometric Gf)
monodromy group of the cover. In dealing with polynomials over finite
fields, if the degree is divisible by the characteristic, the statement
we use for total ramification is that G(Tf, 1). I∞ as a set equals Gf. That is, the cosets by elements in I∞ of the stabilizer of an integer in the representation, G(Tf, 1), cover the group. Group theorists call this a factorization of the representation.
Division of group theory labor:
Once I showed Guralnick and Saxl this factorization property, they were
prepared on the basis of a number of papers extant in the literature to
divide the territory of the classification of finite simple groups, so
as to identify possible factorizations of groups that could be
arithmetic monodromy of an exceptional polynomial.
This is based on the reduction to a primitive Gf,
since the tool for applying the classification is the
Aschbaker-O'nan-Scott theorem that "plugs in" simple groups to get
primitive permutation representations. The category of affine groups
remains separate in this association.
BEYOND THE PAPER:
Almost immediately on publication of the paper, P. Müller, then S.
Cohen, and then H. Lenstra and M. Zieve produced polynomials realizing
those exceptional characteristic 2 and 3 cases. These then, are the
only exceptional (indecomposable) polynomial covers known with
nonsolvable
monodromy groups.
We don't, however, know which affine groups appear as the geometric
monodromy group of exceptional polynomials outside the characteristic 2
and 3 cases.
Corollary 14.2 holds also for an
indecomposable exceptional cover having (at least) one totally ramified
place over a rational point of the base. Its arithmetic monodromy group
is an affine group (if p
> 2 or 3). Again, if q
is odd these
covers are of odd degree.
The methods allow considering covers X
→ P1
–
generalized exceptional covers – that include exceptional polynomials
as a special case. These covers have this property. Over each Fqt
point of the projective line P1,
there is exactly one Fqt
point of X
for infinitely many t.
Thus, X
has a rare diophantine
property when X
has genus greater than 0. It has exactly qt+1
points
in Fqt
for infinitely many t. This gives exceptional covers a
special place in the theory of counting rational points on curves over
finite fields explicitly.
INDECOMPOSABILITY STATEMENT:
Recall this says a polynomial
indecomposable over K
is indecomposable over the algebraic
closure of K.
It holds when the characteristic p
of K
doesn't
divide the degree of the polynomial. Most counterexamples are of p
power degree, but one (due to Peter Mueller) has p exactly dividing
degree of f.
Many applications outside exceptional polynomials benefit
from the Indecomposability Statement. When it holds, it reduces
problems on polynomial values to the case where the geometric monodromy
group is primitive. Indeed, some of technical group theory in [FGS]
appears because we can't easily make such a reduction.