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