Variables
separated polynomials, the genus 0 problem and moduli
spaces
Context of the paper:
This paper consists of a revisiting/updating of the territory of
Fields of definition of function
fields and a problem in
the reducibility of polynomials in two variables, Commun.
Algebra 5,
17–82 (1977). (As of 01/03/08, I still have no
electronic version of
this paper on my web site – like half my papers it was written before
mathematicians started using TeX.)
When I was a 2nd year graduate student in 1965, both Davenport and
Schinzel
were visiting Don Lewis at University of Michigan. The Schur Conjecture, Davenport's problem and problems
about describing the exceptions to the irreducibility of variables
separated polynomials f(x) - g(y)
were a common topic between all three. When f is indecomposable, progress
started on all three with the observation that the last two problems
were – essentially – equivalent.
As an arithmetic geometer, with the emphasis on geometry and
especially ϑ-functions, an unusual aspect of my papers is their
extensive use of finite group theory. Early on, the connection
to the classification of finite simple groups stood out. Later years
are dominated by profinite groups and extensions of simple groups. michyears.html (still in preparation
01/03/08) tells of the things I learned from particular faculty at
University of Michigan. Specifically: Armand Brumer, Don Lewis, Bill
Leveque, Roger Lyndon and John McClaughlin, with each of whom I had
many conversations, informed my later work. Many graduate students, at
Michigan when I arrived in 1964, taught me much. Chuck MacCluer, in
particular, whose thesis on the Schur Conjecture has had a grand
generalization to the theory of exceptional covers (special cases of
Davenport pairs).
The first reducibility problem:
The most telling story at Michigan is my interaction later (after
two years at the Institute for Advanced Study) with Tom Storer, for his
use of cyclotomy. I explain briefly. Davenport's problem bifurcated.
There was a complete simple answer to his question over Q:
If two indecomposable polynomials f,
g over Q have the same value sets modulo
almost all primes p (a
S(trong) D(avenport) p(air)), then g(x)=f(ax+b).
That is, the SDP is trivial.
I approached Storer needing one fact: That -1 was not the multiplier
of a difference set of a doubly-transitive
design. He outlined this, and I put a version of it in the paper. The
result: The polynomials in any nontrivial SDP over
a number field were not defined over R.
This was the first non-trivial use of the B(ranch) C(ycle) L(emma).
Still, it used only elementary group theory. By contrast, the
classification of indecomposable SDPs
over an arbitrary number field – as described in this paper – was a
heavy user of the classification (long before that had been
completed). The most significant point was that the exceptional
polynomials – those with f(x) - g(y)
being non-trivially reducible, with f,
g indecomposable – could only
happen with a limited set of degrees for f (7, 11, 15, 21 and 31).
There are now a growing number of applications for the corresponding
problem over finite fields Fq as recounted in . Even if
you restrict to where the degrees are prime to the characteristic,
starting from an observation of Abhyankar, the paper shows there are
finitely many possible degrees of SDPs.
The methods in characteristic 0 apply to tamely ramified covers, so
these cases show how explicit one can sometimes be with wild
ramification. Indeed, Tony Bluher has made this even more
explicit bluh-Isovalent.pdf.
The second reducibility problem:
A more difficult result in the paper was the culmination of my
understanding of Hilbert's Irreducibility Theorem. A special case of
the particular result: Consider the translates of an
indecomposable polynomial f
(of degree at least 2) over Q:
f(x)-a,
a∈Z. Suppose among these, in addition
to those a assumed by f over Q, infinitely many are reducible.
That is, essentially only the obvious f(x)-a
are reducible (those with a
degree 1 factor). Then, f
has degree 5. Further, the degree 5 case is truely an exception: joint
with Pierre Debes in dfr-deg5.html.
Varying other coefficients are also considered. Peter Mueller, an adept
group theorist, has expanded the problem to consider the problem in far
more generality in his thesis mueller-habil.pdf.
The result was more difficult because the group theory was much harder
– going beyond the classification of doubly transitive finite groups,
and so was the arithmetic – dependent on subtleties in Siegel's famous
theorem on finiteness of integral points on an affine curve. This
problem was the source of my early generalization of Ritt's Theorem: A theorem of Ritt and related diophantine
problems, Crelles J. 264,
(1973), 40–55. My
intention was to point out the large literature on variables
separated polynomials – authors included Davenport, Leveque, Lewis, and
Schinzel among them – where one key question was to classify when f(x)
- g(y) has a genus 0 factor.
My partial
solution was to where the genus 0 factor has at most two places over z=∞, with z defined on the curve {(x,y)|
f(x) - g(y)=0}
as f(x). The papers I quoted mostly took
specific infinite series of separated variables polynomials where it
was common to see that f(x) - g(y)
did not factor. There is a growing literature
and motivation to go in this direction with the works of Avanzi, Bilu,
Couveignes, Gutierrez, Tichy, Mueller, Schinzel and Zieve for which prime-compLaurentPolyPak.pdf
is a convenient reference.
A missed opportunity at Michigan occurred because of the difficulty I
had talking to
Don Higman. His work on modular representations was compatible with
almost every aspect of the development of Modular Towers (my
generalization of the theory of modular curves), though that
development happened only many years later. I got my
understanding of modular representation theory indirectly by
interacting with Serre and his book on Homological Algebra, and reading
Benson's book(s) on Representation Theory.
Topics of the Paper:
1. Historical Introduction 3
1.1. Finiteness results 4
1.2. Moduli space examples and the genus 0 problem 4
1.3. New aspects of irreducibility 5
1.4. Comment on the classification 5
2. Explicit Topics 6
2.1. Variables separated curves as fiber products 6
2.2. Explicit Hilbert’s Irreducibility Theorem for special covers 8
2.3. Synopsis of the group theory handling (2.8) 10
2.4. Frey type Hilbert’s irreducibility problem 12
2.5. [Fri73b] and Rational Points on Variables Separated Polynomials 13
3. Basic Galois Theory Tools for Analyzing Covers 14
3.1. Inertia groups 14
3.2. Tame covers and cyclic inertia groups 14
3.3. Counting points over branch points 15
3.4. Riemann’s Existence Theorem 16
3.5. Grothendieck’s Extension 16
3.6. Remarks on the proof of Thm. 3.3 17
4. Basics on Variables Separated Polynomials 18
4.1. Notations 18
4.2. Basic permutation representation definitions 19
4.3. Translating curve properties to group theory 19
4.4. Reduction to equal splitting fields 19
4.5. Newly reducible pairs 20
4.6. Applying RET 20
4.7. Drawing conclusions from Lemma 4.3 21
5. Positive Characteristic—Loss of Riemann’s Existence Theorem 21
5.1. Simple groups as a resource for investigations into affine groups
21
5.2. Producing monodromy groups in PGLm+1(Fq)
22
5.3. Examples ofAbhyankar 23
5.4. Part of Thm. 5.2 characterizing (5.4c) 23
5.5. Davenport’s problem in positive characteristic 26
6. The genus 0 problem and its relation to this paper 27
6.1. Resolution of the original conjecture 27
6.2. More precise expectations 28
6.3. Appearance of generic curves of genus g 28
6.4. Qualitative implications of the genus 0 problem 29
7. Parameter spaces for arithmetically related covers 31
7.1. Prelude on components of families 31
7.2. For the Inverse Galois Problem: G
= A5, C = C34
32
7.3. Expectations of the universal parameter space of such covers 32
7.4. Representatives of absolute Nielsen classes 33
7.5. Groups Br, Hr and Mr
33
7.6. Modulo PSL2 action 33
7.7. Special generators of π1(P1j \
{0, 1,∞}) 34
7.8. Branch cycle description of ¯Hrd → P1j 34
7.9. Showing ¯Hrd
= P1w
over Q 34
7.10. Branch cycle description of ¯Hin,rd → P1j 35
7.11. Adding the branch points 36
8. Degree 13 Davenport Polynomials 37
8.1. Group theory setup for Davenport polynomials of degree 13 37
8.2. Listing elements of a Nielsen class 38
8.3. Conclusions from the computation of Q1 and Q2 39
8.4. Defining field for H({1,
2, 4, 10}) 39
8.5. Rationality of H({1, 2,
4, 10}) 40
9. Using the Classification 40
9.1. Proof of Thm. 9.1 41
9.2. Exceptional groups 41
10. Polynomials with Frey-type Irreducibility 42
10.1. The covering space setup 42
10.2. Applying Faltings’ Theorem 43
10.3. Finding f satisfying
Question 10.1 conclusion 44
11. Problems from Variables Separated Polynomials 45
11.1. Comments on [BST99] 45
11.2. Comments on [Haj98] and [Haj97] 46
11.3. Reducibility of f(x) − h(y)
when f is composite 46
11.4. The (n,m) problem 47
11.5. Mueller’s results extending Davenport’s problem 48
References 48