Variables Separated Equations and Finite Simple Groups

This is a mathematically more complete version of UMStoryShort.html that came eventually from the courses/conversations when I was a graduate student at University of Michigan ('64-'67). The initiating problem is from Harold Davenport, who – with Don Lewis and Andrzej Schinzel – wrote several papers on the behavior of finite fields under polynomial maps.  Many unsolved problems of that time, most unrelated to finite fields, featured separated variable equations. This included Schur's Conjecture from 1921, with its forerunners going back to Hermite, and the middle 1800s.

I was looking for a thesis challenge that would combine function theory and group theory, and allow me to grow without pinning me to one discipline. This report features the transitions between three disciplines:
  1. variables separated equations (special algebraic equations);
  2. group (and Galois) theory; and
  3. complex variables (as formulated by Riemann).
I first noted that problems on variables separated equations asked for relations between algebraic covers by genus 0 curves. I used that to apply group  theory. The function theory arises in inverting group theory to form equations. The monodromy method, developed to solve Davenport's Problem, also helped others solve related problems. UMStoryCoeffics-Equats.html and UMStoryExc-OIT.html  summarize some of those using transitions – from Davenport's problem – between areas a, b, and c. 

Algebraic equations occur in many modern data problems. They represent relations between variables defining data. Technically, the data variable gives us a monodromy (or Galois) group with a faithful permutation representation.  Experts regard data-variable problems as Diophantine. They should have convenient coefficients, such as ordinary fractions, Q. In that case, there are two monodromy groups: the arithmetic (over Q) and a normal subgroup of it, the geometric (over the algebraic closure).

There is an encompassing inverse problem: Given such a pair of groups (with their compatible permutation representations), one normal in the other,  find an equation and data-variable over Q, having that the arithmetic, geometric monodromy group pair.

Also, the data-variable has many uses. Example: exceptional covers for Cryptography: Over infinitely many prime residue classes, their data-variable maps one-one from the data to its values. Many renown problems, such as Serre's "Open Image Theorem," translate to classifying types of exceptional covers. The Schur Conjecture was the proposed classification of such covers where the data variable is a polynomial (on a genus zero curve).  

Davenport's problem was to essentially classify polynomials over Q by their ranges on almost all residue class fields. The most general results, restricted to polynomials not composable (indecomposable) from lower degree polynomials, gave two very different conclusions:

Over Q two polynomials with the same range were linearly equivalent: obtainable, one from the other, by a linear change of variables.
For certain number fields polynomials that weren't linearly equivalent could have the same ranges for all residue class fields, though the exceptions cases were understandable and limited.

Schur's original conjecture was technically much easier than Davenport's Problem. Still, by considering its analog for rational functions, the monodromy method connected to Serre's O(pen)I(mage)T(heorem)  (UMStoryExc-OIT.html) and, so, to modular curves. My summary of that starts with a Davenport-Lewis paper.

The monodromy method included two new tools for investigating algebraic equations: the B(ranch)C(ycle)L(emma) and the Hurwitz monodromy group. By walking through Davenport's problem with hindsight, we see why the – rarely acknowledged – preoccupation with variables separated equations gave important lessons on these tools, and more generally, an approach for those unacquainted with groups to using them.

Davenport's Problem was more explicit with these tools than in their general application [Fr77], though, without question, they arose here seriously.  That can be seen by it's contribution to the most general problem from the heart of the monodromy method: What we call the Genus 0 Problem. We celebrate the place of these pieces in function theory results by calling attention to these lessons: 

What allows us to produce branch cycles §VI.4.
What is in the kernel of the Chow motive map §VII.2.
What 'in nature' (a phrase from [So01], see §I.3) gives today's challenges to group theory  §VII.3.

Each phrase addresses an aspect of formulating problems based on equations. That is, many disciplines seem to need algebraic equations. Yet why, and how much do we lose in using more easily manipulated surrogates for them? This exposition ties together disparate observations forced on me by having to put results in different publications (I explain that below). It also adds points relating theory to the whole enterprise of writing explicit equations.

Continuing an Abel-Galois-Riemann (before 1872) tradition, the method made contributions to the existence problem. For any problem in this domain, algebraic covers fall in continuous (connected) families. Often, then, solutions of a problem follow from identifying the piece(s) – reduced Hurwitz space components – where desired solutions fall.  Even Abel, Galois and Riemann would have been surprised at some of the explicit consequences of Davenport's problem: Especially: The use made of the classification of finite simple groups, and separately, how it led to the formulation of the genus 0 problem.


Table of Contents

I. The relation between four problems:
I.1. Introduction to Davenport's Problem:
I.2. Detecting a 'few' exceptions:
I.3. The "Genus 0 Problem:"
I.4. The Context for UM affiliated faculty:
II. Separated variables equations and group theory:
II.1. The effect of splitting the variables:
II.2. Formulations between the 1920's and the 1960's:
II.3. Galois Theory and Fiber Products:
III. Moving from Chebotarev translation to Riemann Surfaces:
III.1. My Choice of Thesis Topic:
III.2. A Version of Chebotarev's Theorem and Seeking a Converse:
III.3. Meeting UM Faculty:
III.4. Going to ∞:
III.5. Combining data at ∞ with Chebotarev:
IV. Distinguishing between doubly transitivity and primitivity:
IV.1. Translating Primitivity:
IV.2. Group Theory in Graduate School:
V. Properties of equations without writing equations:
V.1. A linear relation from Davenport's hypothesis:
V.2. Different Sets and a Classical Pairing:
V.3. Misera's example (sic):
V.4. Group theory immediately after Graduate School:
VI. The B(ranch)C(ycle)L(emma) and Solving Davenport's Problem:
VI.1.  The action of GQ:
VI.2. Applying the BCL to Davenport's Problem:
VI.3. Producing Davenport pairs:
VI.4. Branch cycles, the tie to group theory:
VI.5.  Computing and Using a Nielsen class:
VII. The significance of Davenport's Problem:
VII.1. The Genus 0 Problem:
VII.2. Attaching a zeta function to a diophantine problem:
VII.3. Applied group theory and todays challenges occuring 'in nature:'
VII.4. Final UM Comments:

Appendix on Group Theory and Branch Cycles
§ App.1. Some useful group definitions:
§ App.2. How could I possibly have understood the group theory in [FrGS]?:
§ App.3. Branch cycles produce an algebraic cover:
§ App.4. Grabbing a cover by its branch points and braids:
§ App.5. Three genus 0 families of Davenport Pairs:
Bibliography

I. The relation between four problems:

Davenport stated his problem at a conference at Ohio State during my 2nd year of graduate school. The anchors for this story are the result I proved and the problem that "seized" John Thompson – his own word – that came from it. I call the former D1 and the latter G1(0) below.

I.1. Introduction to Davenport's Problem: D1 said that two polynomials f and g over Q with the same ranges on almost all finite fields, with f indecomposable, must be related by an inner change of variables α, f(α(x))=g(x), α=ax+b an affine transformation, or f and g are affine equivalent. Actually, Davenport didn't include the hypothesis of indecomposable, though for progress on his problem this was essential (see Müller's Conj).

Note that if f and g are a pair of rational functions that have the same ranges for almost all primes p, then so will αof and αog, their outer composition with an affine transformation (if the transformation has coeffficients in Q). If you compose f with both inner and outer Möbius transformations, we say the result is Möbius equivalent to f.

Problem G1(0) said that all genus 0 covers have special covering (monodromy) groups (§I.3). Indeed,  this has two parts: The genus 0 monodromy groups that come in large families, and exceptional genus 0 monodromy. Problem G1(0) became known as "The genus 0 Problem."  

The archetype use of the word 'exceptional' first appeared in the solution of Davenport's problem over an arbitrary number field, the problem I call D2. All higher rank projective linear groups – examples of almost simple groups (App. 1) – over finite fields might have yielded solutions countering the expected Davenport conclusion. Yet, function theory showed only finitely many gave solutions of D2. Further, many of the most striking exceptional genus 0 monodromy groups appeared either from Davenport's problem, or from genus 0 upper half plane quotients that are 'close-to' modular curves. Those from problem G2(0) (§ I.2). 

Abel and then, Galois – aware of long monographs by Lagrange and his students – inspected particular equations to conclude that precise group theory could solve long-standing equation problems. Here is one quick summary of much early 1800s mathematics. Galois showed the impossibility of uniformizing the function fields of the modular curves X0(p) (introduced by Abel), p a prime ≥ 5,  by radicals.

The 20th century didn't much use the phrase "uniformized by radicals." Yet, despite attempts to avoid such an old formulation, a variant of it dominated published results in the 1960s. The algebraic equations I heard most about in graduate school had separated variables:

(*) f(x)-g(y) = 0 with f and g polynomials, whose degrees we take (respectively) to be m and n.

An initial small step, introducing a pair of covers of the Riemann sphere, opens the territory to using group theory. Rewrite (*) by introducing z, so as to split the variables:

(*2)       f(x)-z = 0 and g(y)-z = 0.

Affine equivalence of pairs: Questions on solutions of (*), when interpreted in (*2) form are equivalent to those with (αof (α'(x)), αog(α''(y))) replacing (f(x),g(y)), with α, α' and α'' affine transformations. We say the former pair is affine equivalent to the latter.

Using  (*2) interprets (*) as relating two genus 0 covers (§II.1).

Is it surprising that there are still mysteries about genus 0 covers? We have as a subtheme being precise about the most jarring ingredient from R(iemann)'s E(xistence) T(heorem, §VI.4). That is, how covers of the Riemann sphere relate to branch cycles.  When the covers have genus 0 and especially when they appear naturally, many feel uncomfortable – as did Kronecker and Weierstrass – without an explicit uniformization. Handling that discomfort appears often in papers to which UMStoryCoeffics-Equats.html and UMStoryExc-OIT.html refer.

I.2. Detecting a 'few' exceptions: The major surprise in Davenport's Problem was that D2 was almost – for all but finitely many degrees, but definitely not for all – true. Young mathematicians can founder on the intuition that simply stated conjectures should be true. When they aren't, detecting why, and when not, usually calls for new ideas. An alternative, of course, is to abandon the problem.

The explication and presentation to the community of D2 – its finitely many exceptional degrees – gave three results relating finite group theory to algebraic equations. [Fr80] emphasized the connection between these problems and the (finite) simple group classification.

I had little official group theory or arithmetic geometry background. Still, I had found tools that related the two areas, to the advantage of both. My  unofficial background, from many affiliated with the UM mathematics department, made this possible. It took time for me to appreciate this and how much it prepared me to persist and learn new topics. The profession does not much encourage learning new topics.

A year lapsed between graduate school and the conversation with Tom Storer – during the summer of 1968. I needed that year to be certain of the distinction between D1and D2. Also, Thompson's name was attached to another genus 0 problem, which I call G2(0). This said that the j-line covers appearing in G2(0) should have explicit uniformization by (upper half-plane) automorphic functions attached to representations of the Monster Simple group. They called it "Monstrous Moonshine" and its resolution won Borchards a Fields Medal.

I first heard of G2(0) during the group theory conference called Santa Cruz. (Its proceedings included my exposition [Fr80].) Even more time elapsed between the paper purporting to connect G1(0) and G2(0). The greatest accident must be the conversation between Thompson and I, while walking to lunch not long after my arrival at U. of Florida (§ VII.1).

I.3. The "Genus 0 Problem:" Genus0-Prob.html has the precise statement of the Genus 0 Problem. A minimal rough statement is that, due to the nature of branch cycles (§VI.4), monodromy groups of rational functions fall – with rare but significant exception – among groups that most mathematicians have heard about. Those exceptions, as in Davenport's Problem, have a serious impact.  The Genus 0 Problem formulation, and much work on it, is due to Bob Guralnick.

Separated variable equations are a useful archetype in the theory of Riemann surfaces, starting with hyperelliptic curves where Riemann first proved a generalization of Abel's Theorem. The Genus 0 Problem is a key step in considering what attributes of these equations qualify as special.
 
From the solution of D2, three genus 0 curves, each a natural upper half-plane quotient and j-line cover, though not a modular curve, arise as parameter spaces for Davenport pairs. They appear in other problems, too. From §App.5 (for n = 7, 11, 13),  each space has an attached group representation of a projective linear group.  Does this lead to an explicit automorphic uniformizer of each?

A group theorist does demandingly intricate work. What I, not a group theorist, found is that applying groups to the existence of algebraic relations brought you quickly to its depth. Especially: you can accomplish goals by seeking moments when group theory can circumvent tiresome equation manipulations. Getting a balanced education was likely forced on me in going between two such different places as UM and IAS/Princeton. More so, when I decided I was unwilling to drop the best – for an algebraist with a complex analytic bent – of either.

So much of modern group theory has little to do with permutation representations, despite that so much of group theory's birth does. Indeed, while [So01, p. 315–317] does give a view of the birth of group theory, even in its referral to Galois it differs big-time from my view. An audience question to Ron Solomon, when he gave on his history lecture [So01] at UF, was [roughly], "How did Galois' work survive?" I suggested then an elaboration of the path through Jacobi's interest in the uniformization result mentioned in § I.1. I also mentioned that savior to both Abel and Galois, Crelle, before it got to the renovation and update by (yes, brilliant and crucial to Galois' survival) Jordan.

Yes, there is some agreement between this story and [So01, p. 347]: "... experience shows that most finite groups which occur 'in nature' – in the broad sense not simply of chemistry and physics, but of number theory, topology, combinatorics, etc. – are 'close' either to simple groups or to groups such as dihedral groups [I would have said appropriate affine groups as in the § App.1], Heisenberg groups, etc, which arise naturally in the study of simple groups." The agreement is that a researcher requires some handle on simple groups to get going.  Still,  §VII.3 exclaims that applied group theory has challenges that  put the present approach to simple groups in a new perspective.

I.4. The Context for UM affiliated faculty: Davenport was part of a school that didn't much care for group theory (or Galois theory). He would have had a spasm at my viewing his problem "motivically" as in §VII.2. Still, the reach of such an elementary problem, it's relation to so much literature, and it raising such questions about variables separated equations, recommends it as contribution to finding meaning in Solomon's phrase 'in nature.'

With a superscript "a" for visiting junior faculty, "v" for visiting senior faculty, and "s" for (fellow) student, this  is a review of how the mathematicians A. Brumera, R. Bumbya, H. Davenportv, D.J. Lewis, W. Leveque, R. Lyndon, C. MacCluers, R. Miseras, J. Mclaughlin, A. Schinzelv, J. Smitha gave me images of how to connect disparate areas into a coherent story. These people were associated with University of Michigan during my three years – 1964–67 – of graduate school. Later T. Storer played a crucial role. The names J. Ax, R. Brauer, W. Feit, R. Guralnick, P. Mueller, J. Saxl, J.P. Serre, G. Shimura and J. Thompson appear substantively, too. It is difficult to separate the two years following, and then the subsequent rearising of the topics from those graduate school years.

I have heard many preach such received wisdom as "achievement – in mathematics – will bring success." To me that gives new meaning to the lives of Abel, Galois and Riemann, all poorly recognized achievers in their lifetimes. § VII.4 addresses just one issue related to "success vs achievement" from my time at UM.

II. Separated variables equations and group theory:

The effect of using form (*2) in place of (*) is to consider the relation between two covers by Riemann spheres, f: P1x→ P1z and g: P1y→  P1z of the Riemann sphere P1z. Recall: P1z  is just projective 1-space, but with the variable z indicating an explicit isomorphism with affine 1-space union a point, ∞, at infinity. We always assume both f and g are nonconstant.

II.1. The effect of splitting the variables: Equation (*) defines an algebraic curve in affine 2-space. It has a completion in projective 2-space, with homogeneus variables (x,y,w), by forming the curve wu(f(x/w)-g(y/w))=0, u the maximum of m and n. This isn't, however, the unique projective nonsingular algebraic curve completion of the affine curve, for it may have singularities.

One immediate advantage of using (*2) is to geometrically describe such singularities. They correspond to the pairs (x',y') that both ramify in the respective maps f and g to P1z. That is, regard (*) as the fiber product – set of pairs (x',y') with f(x')=g(y') – of the two maps f and g, but extend the fiber product over ∞. papers use the notation P1x×P1zP1y. We call any z value over which there is a ramified point on P1x a branch point of f. Note: If f = g (and m > 1), then the fiber product has at least two components, one the diagonal.

Then, P1x×P1zP1y is projective, too, a closed subset of P1x×P1y. Still, this contains (*) as a subset, so might be singular. In Davenport's Problem this applies seriously. The reduction DS1 considers f and g (not affine equivalent), yet n=m and  f and g have exactly the same branch points.  In our discussion we alway want a nonsingular projective model of (*), which maps naturally to P1x×P1zP1y, and is a one-one map (immersion) except over the singular points. (In higher dimensions, the right object is the projective normalization of the variety.) 

[chpfund.pdf: §3.3.2, §4.2.2 and §4.3] discuss these compactifications in much more detail, including elaborating on the following remarks.
The issue of "explicit equations" for algebraic sets related to separated variables comes up repeatedly in UMStoryCoeffics-Equats.html. Having explicit separated coordinates, is part of their charm. Yet, as with modular curves, birational equations don't hand you equations for their unique normalization.

II.2. Formulations between the 1920's and the 1960's: Concentrating on equations from (*) (sometimes f and g are rational functions), and combining it with questions about solutions, say in the rationals Q, explains many  papers of that time.  Here are examples fitting this paradigm I heard from Davenport, Leveque, Lewis and Schinzel my second year of graduate school. All assumed f and g had coefficients in Q: Z/p refers to the integers modulo a prime p.
  1. Which equations (*) have infinitely many solutions in Z (or Q)?
  2. Schur (1921): If f(x)=x, which equations (*) satisfy this for infinitely many primes p: For each x' ∈Z/p  there is  y' ∈Z/p satisfying (*).
  3. Davenport (1966, at a conference at Ohio State): Which equations  (*)  have this property for almost all primes p:  For each x' ∈Z/p  (resp. y' ∈Z/p) there is y' ∈Z/p (resp. x' ∈Z/p) satisfying (*): f and g have the same values mod p for almost all p.
  4. Schinzel (papers from the late '50s): Which equations (*) factor into lower degree polynomials in x and y [Sch71]?
In referring to these below, I will always assume the hypotheses hold nontrivially. For example: exclude g(y)=f(ax+b) in Davenport's problem, for then the conclusion to his question obviously yes if a, b are in Q. Refer to a nontrivial pair (f,g) satisfying #3 as a Davenport pair (over Q). Using almost all residue class fields of a number field K gives meaning to a Davenport pair over K.

In addition to the problems above, a H(ilbert)'s I(rreducibility) T(heorem) variant kept appearing. Archetypal of problems unsolved at the time was this:

(*3) For which f are there infinitely many z' ∈Z where f(x) - z' factors over Q, but it has no zero in Q?

Most papers of the time considered special polynomials f and g, answering some variant on these problems negatively. Example: For f in some specific set of polynomials, the answer to #2 would be that none had Schur's property.

In some ways the essence of algebraic equations, in two variables, is caught by the isomorphism class of the equation, represented by a point on the moduli space of curves of a given genus. In other ways, not, for that doesn't hint at the relations (correspondences) between equations.

Further, those equations that -- with a change of variables -- have an expression with coefficients in the algebraic numbers, maybe even in Q, differ extremely from those that do not. It is possible to look at such equations combinatorially -- say with zeta functions -- by asking about their behavior when the variables assume values in almost all finite fields. The theory of Chow motives approaches algebraic equations in great generality.

Davenport's problem, as I will show, led by specific problems to another view of some of the same issues. Especially significant, was that over certain number fields there were Davenport pairs in great abundance. That is, they formed a nontrivial family of such pairs. In depicting that family, especially in describing efficient parameters for it, I was running up against the total lack on the part of most algebraists for a description of any moduli problem. It was the analog of asking someone the meaning of solving equations: If they knew how to solve them, then giving a solution was the meaning. If, however, they did not know already how to solve them – in this case, what were the appropriate parameters to display them – the problem seemed meaningless to them.

Most vexing to me, was that I was showing how to find the parameters, based on combining group theory and analytic continuation, yet there was no recognition that this was a viable approach. § App.5 recounts the three families of davenport pairs, corresponding to the degrees 7, 13 and 15, and the different equivalences on them that produced parameters for them to solve problems. These parameter spaces each have a genus 0 curve at their core. I snuck them into papers to illustrate various phenomena, and here explain conclusions from those phenomena in one place.

II.3. Galois Theory and Fiber Products: Groups appeared little in §I.2. problems up to 1967. Yet, progress came quickly after introducing them. Here is how they enter. For simplicity assume f and g over Q. Each of the maps f: P1x→ P1z. and g: P1y→ P1z has a Galois closure cover over Q, 
ˆf: ˆX   →P1x and ˆg: ˆY →P1y.
Abel and Galois would recognize these (at least over the complexes). So, they have Galois groups aGf and aGg, the automorphism groups of these covers. Indeed, the Galois closure of f is naturally (normalization of) any connected component (over Q) of the m-fold fiber product of f minus the (fat) diagonal components [chpfund.pdf, §7.3].

The small "a" at the left stands for a(rithmetic), and indicates this complication:  In situations like this, where the polynomials f and g are far from general,  an irreducible component (over the complexes) of the cover ˆX has equations over a field  ˆQf, larger than Q.

It was standard in the literature of the time to assume ˆQf = Q. Significantly in the general problems I faced, it wasn't. Especially in Problem §II.2 #2, and the connection of that problem to one of Serre's {sl Open Image Theorems}.

When a student comes upon such things, there are decisions about whether to ignore it or not. I did not, though it took time to convince others this was necessary.

There is also a minimal Galois cover of P1z that factors through both ˆX and ˆY. Its group, aGf,g, is naturally a fiber product. Indeed, define ^W to be the largest (nonsingular) Galois cover of P1z, over Q, through which both ^f and ^g factor. So, there is ˆfw: ˆX → ˆW and ˆgw: ˆY → ˆW factoring through the maps to P1z. Each automorphism σ of ˆX or ˆY  induces an automorphism rσ of ˆW. (The superscript "r" stands for restriction.)

Then, aGf,g is the fiber product, {(σ1,σ2) ∈aGf×aGg| rσ1= rσ2 on ˆW}. Further, with m and n the respective degrees of f and g, then aGf,g naturally has permutation representations Tf and Tg of degree m and n.

III. From Chebotarev translation to Riemann Surfaces, growing as a student:

Brumer taught me Algebraic Number Theory during a time Lewis was in England, the Fall semester of my 2nd year. Brumer attended a course by McLaughlin on group theory, and further, he included comments on groups during our private black board discussions.

III.1. My Choice of Thesis Topic: In Brumer's course I learned the fiber product construction of the group of the composite of two Galois extensions of a field. He also taught the standard Cebotarev density theorem, and gaves problems on using groups to interpret it.

During an algebraic curve course (Spring 1966) my thesis topic congealed. Suppose we have a collection of polynomials g1, …, gt.

(*4) What was true of f whose range on Z/p for almost all primes p is in the union of the ranges of Z/p of g1, …, gt?

Polynomials give algebraic maps. The first distinguishing property of a polynomial is its degree, which one can see quite plainly. Another property is subtle enough that it may not come to mind to those inexperienced. Even to those with experience, it's not obvious what to do with it, for it can seem a jump into great complication. I speak of the monodromy group.

A generalization of (*2) would consider aGf,g1,…,gt, the Galois group of a cover formed from many fiber products, with permutation representations Tf and Tg1, …,Tgt of respective degrees m and n1,…, nt.

III.2. A Version of Chebotarev's Theorem and Seeking a Converse: The Chebotarev density theorem translates (*4) into a statement on aGf,g1,…,gt.   Suppose a complex component of the cover whose group is aGf,g1,…,gt has definition field ^Qf,g.  Then, aGf,g1,…,gt maps surjectively to the Galois group G(^Qf,g/Q).  The kernel is the geometric Galois group, Gf,g1,…,gt, of the cover.

For τ ∈ G(^Qf,g/Q), the elements of aGf,g1,…,gt mapping to τ form a coset, τaGf,g1,…,gt in the arithmetic group. So, if statement  (*4) holds for infinitely many primes rather than almost all, this would imply:

(*5) For some coset τaGf,g1,…,gt and for each σ ∈ τaGf,g1,…,gt, Tf(σ) fixes a letter if and only if for some i, Tgi(σ) does.

This statement is a combination of Cebotarev – actually not then in the literature [Fr76, p. 212-13] –  for number fields and for function fields. A much subtler point is that (*4) and (*5) are equivalent according to the generalization of MacCluer's Theorem (see UMStoryExc-OIT.html).

If the statement is about all primes – as in Davenport's problem – then,  it translates to this simpler seeming statement: 

(*6) For each  σ ∈ aGf,g1,…,gt, Tf(σ) fixes a letter if and only if for some i, Tgi(σ) does. 
C1: What a geometric converse would be: Statement (*6) has a 100% group theory version. A converse would ask, if given any group statement of this ilk, are there (f,g) that produce the group conditions.

C2: What an arithmetic converse would be: Statement (*5) has a group theory version about an arithmetic monodromy group. A converse would ask if, given two groups aG and G satisfying a statement like (*5), if there are covers that realize these two groups as their arithmetic/geometric monodromy groups over some number field.

Here are subtler points on C1 and C2.
III.3. Meeting UM Faculty: The graduate student population was over 200 at UM in those years. I later realized that the department was large, too, compared to other department in which I ever held a position. Therefore, seminars often started with many attending, yet dropping rapidly each week.

At the time, compared to the best of graduate students at UM, I started with a meager mathematics background. I had been an undergraduate Electrical Engineer (albeit having finished in two years), followed by three years as an aerospace engineer, first in Boston, then in my hometown of Buffalo.

Characteristic of me, I stuck actively with most seminars to the end.

I early learned fiber products at UM because we had a seminar on Diudonne's version of Grothendieck's writing called EGA, summer 1965. About 50 people showed at first, but later there was just Brumer, Bumby and me. Brumer later said the seminar was just me, though I recall practicing sheaves, direct limits and projective limits especially from a famous Grothendieck paper – called Tohoku – under their tutelage. Bumby gently guided me to an intuition on direct and inverse systems, and much profinite homological algebra. That came in handy throughout my career.

Lewis arranged for my attendence at two Bowdoin college NSF-funded summers. Eight weeks each on Algebraic Number Theory (summer of 1966) and Algebraic Geometry (summer of 1967).  Both summers I learned everything put in front of me.  I also learned I would be regarded as an ignoramus for not having the background prevalent then at Harvard, MIT or Princeton. My remedy: pick that up, too, especially the algebraic geometry.

Brumer left for Columbia at the start of my 3rd year. Imitating Brumer, I engaged McLaughlin directly in blackboard discussions when I could catch him, about permutation representations. Another handy seminar was run by Roger Lyndon and me: Discontinuous groups acting on the upper half plane. At the same time, I read notes of Brumer on modular curves from lectures of Gunning. As with theta functions, this became a hidden tool for me, ready for action when necessary, augmented sharply by the two years I spent around Shimura while I was at the I(nstitute for)A(dvanced)S(tudy) 1967–1969.

III.4. Going to ∞: I was well aware, at the end of Spring 1966 that coming to (*6) was no big accomplishment for these reasons:
  1. It said nothing about the polynomials involved, not even suggesting what, of significance, one might say.
  2. It had only a mild connection to the serious tools of mathematics.
  3. The problem didn't register with the MIT-Princeton-Harvard students at the 1966 Bowdoin Summer.
Sometimes elementary observations open up problems. If some famous algebraic geometer had cued my next step – say Artin or Mumford, both of whom I knew well later – it wouldn't have resonated as the big step it was. To see it, however, through a graduate students' eyes broke me into a new way of thinking.

Later I realized it was a stride even for Riemann. Lefschetz admitted he finally understood Picard from something similar. Yet, doesn't this sound elementary?: I looked at ∞,  Christmas morning 1966, at a time I despaired at there even being any structure to problem (*4).

What came to me was a finger circling ∞ on the Riemann sphere, clockwise (so, unlike most people I put my loops clockwise around points to this day), and then coming back to a basepoint – at my feet. Here's what it meant for understanding the values of a polynomial f:  P1x → P1z. You knew for certain one element, σ∞, in Gf (and so in aGf): an n-cycle coming from the cover totally ramifying over ∞.  Recall, ∞ was not initially included in the values of  f, but that is irrelevant. 

III.5. Combining data at ∞ with Chebotarev:  That same finger circling ∞ corresponded to a path, on the punctured sphere, and so to a generator, σ∞, for the inertia group over ∞ for the fiber product of all covers given by f and the gis. In each corresponding permutation representation σ∞ would appear respectively as an m-cycle or an ni-cycle.

Conclude from Chebotarev in (*6): With N the least common multiple of the nis, m divides N.

Here is why: The element σ∞N fixes every letter in Tgi (corresponding to gi). So, from  (*5), it must fix at least one letter in the permutation rep. Tf. Yet, unless m divides N, as Tf(σ∞N) is an m-cycle to the N-th power, it fixes none of those letters.

For example: In Davenport's problem (#2 in § II.2), this immediately implies the degrees of a Davenport pair ( f, g) must be the same.  (From here on we take this common degree as n.) In fact, there is a strong conclusion.

DS1: Suppose f and g nontrivially satisfy Davenport's hypothesis. Then their Galois closure covers are the same [Fr73, Lem. 6].

IV. Distinguishing between doubly transitivity and primitivity:

Unless you are a group theorist, or have – through some particular problem – met groups more than the typical mathematician, then you likely know finite groups only through their permutation representations. Even then, you are unlikely to realize that there is an intimate relation between primitive groups and simple groups (§ App.2) – excluding primitive affine groups (§App. 1), which may resist any classification. Further, those that are doubly transitive are more familiar, maybe even easier.

I didn't know these things, partly from [A-O-S], when I started either. I luckily skirted along the easier edge of the doubly transitive/primitive divide in these problems. 

Even today, after 35 years of evidence that all simple groups have been listed, primitivity still causes problems. Moreso, if you can't assume a permutation group is primitive, even the classification has yet to be helpful (§ VII.3 and UMStoryCoeffics-Equats.html).

IV.1. Translating Primitivity: The monodromy group aGf of a cover f: X → P1z over a field K is primitive if and only if the cover does not trivially factor  through another cover (over K). It is doubly transitive if and only if the fiber product X×P1zX has exactly two irreducible components (over K, one of which must be the diagonal). For f a rational function, primitive means f  doesn't decompose as f1o f2 with both fis of degree exceeding 1 (over K).  Doubly transitive means  (f(x)-f(y)/(x-y) is irreducible (as a polynomial in two variables) over K. 

Given a permutation group G, acting on {1,…,n}, denote its subgroup of elements fixing 1 by G(1). Galois theory translates these respective statements as conditions on aGf under the permutation representation Tf.
If Gf is primitive, then so is aGf, but the converse does not in general hold. Still, we have the following – essentially my first mathematics lemma.

Polynomial Primitivity:  If f, a polynomial, decomposes over an algebraic closure of K, then it decomposes over K.

In Schur's Conjecture primitivity is very helpful. A composite of polynomials gives a one-one map on a finite field, if and only if each does. Polynomial Primitivity allows reverting to where G f (the geometric group) is primitive. Further, two famous group theory results from early in the 20th century help immensely.
IV.2. Group Theory in Graduate School: Richard Misera, a fellow graduate student – I never saw him again after getting my degree – was studying with Don Higman. He volunteered an example after he saw me discussing distinction between permutation representations and group representations with McLaughlin. That example became a powerful partner in my quest to solve Davenport's Problem (§ V.3) though I knew it only from my conversation with Richard, and long before the rest fell into place.

Soon after graduate school, I knew enough to solve Schur's Conjecture (§ I.2). Still, it was John Smith, whom I thought I saw by accident at IAS – when he actually came to discuss a problem with me – who told me of Schur's and Burnside's Theorems. Smith was the 3rd  (and last, including MacRae and Schinzel) person who was at Michigan  during my graduate years with whom I wrote papers (in each case two).

My Erdös number is 2 because of Schinzel.

V. Properties of equations without writing equations:

In contrast to many algebraic equation papers, even those using the monodromy method, Davenport's problem was solved mostly on general principles, without extensive equation manipulation.

V.1. A linear relation from Davenport's hypothesis:

DS2: Suppose f and g nontrivially satisfy Davenport's hypothesis, and f is indecomposable.
  1. Then, Tf and Tg are inequivalent as permutation representations of aGf = aGg. Yet, they are equivalent as group representations.
  2. Further, the converse holds: Such permutation representations, equivalent as representations, imply f and g satisfy Davenport's hypothesis (MacCluer's Thm. in UMStoryExc-OIT.html), and (for almost all primes p) f and g assume each value mod p with exactly the same multiplicity.
  3. Finally, because f is indecomposable, so is g and the representation statement is equivalent to f(x)-g(y) being reducible (Shinzel's problem, §II.2).
Result #1 in DS2 has a field theory statement that uses some geometry from the maps. Suppose f and g have definition field K. List the zeros, xi (resp. yi), i=1,…,n, of f(x)-z (resp. g(y)-z) in an algebraic closure of K(z) Also, do a penultimate normalization: change x to x+b, b ∈K, so the coefficient of xm-1 is 0 (similarly for g).

DS1 says K(xi, i=1,…,n)= K(yi, i=1,…,n). So, xi is a rational function in the yjs. What DS2 says is that xi is a sum of distinct yjs times a nonzero element a ∈K. With no loss, take a=1, and write

(*7) x1=y1+ yα2+…+yαk, with 2 ≤ k ≤ (n-1)/2 (because the complementary sum of yis now works as well).

Result #3 is clearly geometric. A general preliminary statement goes like this [Fr74, Prop. 2]: If f(x) and g(y) are rational functions over a field K (assume K has characteristic 0, or that the covers given by f and g are separable), then we can write f (resp. g) as f1o f2 (resp. g1o g2 ) so the following hold.
For rational functions, however, there is no conclusion like (*7)  without that n-cycle. You can't even say the degrees of  f1 and g1are the same. Classifying variables separated factorizations was Schinzel's Problem, not Davenport's. Their mathematical common ground seems to have been built around variables separated equations.

They had not considered the equivalence of their problems for the case f  is an indecomposable polynomial. They aren't equivalent without the indecomposable assumption. All attempts to write equations for Davenport pairs used Schinzel's factorization condition.  UMStoryCoeffics-Equats.html has more on the rational function case.

V.2. Different Sets and a Classical Pairing: People who like cyclotomy (Gauss did, and so did Davenport, for example) see difference sets in many situations. The kind that arises in this problem is special (cyclic),  though it is an archetype. Denote the letters of Tf (resp. Tg) by xi (resp. yi), i=1,…,n.

Normalize the naming of the letters in each of the permutation representations so that σ∞ (§III.5) cycles the xis (and the yjs) according to their subscripts. Combine double transitivity and the action of σ∞ on both sides of (*7) to see from where comes the definition of difference set.

DS3: The collection of integers R1={1,α2,…,αk} mod n has among its nonzero differences each integer 1,…, n-1, exactly u=k(k-1)/(n-1) times.  Further, writing the yis as expressions in the xjs gives the attached different set (up to translation) as R1 multiplied by -1.

Argument: Acting by σ∞ on R1 – translating subscripts – gives a collection Ri,  i=1,…,n. The permutation action of Gf gives a representation equivalent to Tf. The number of times an integer u mod n appears as a (nonzero) difference from R1 is the same as the number of times the pair {1, u+1} appears in the union of the Ris. That is, you are normalizing its appearance as a difference where the first integer is a 1. Double transitivity of Gf is equivalent to Gf(1) is transitive on 2,…,n. So, there the count of the appearances of {1, u+1} in all the Ris is independent of u.

Now consider, as in the last sentence, writing the yis in terms of the xjs. To do so consider a classical n×n incidence matrix: Ix,y: rows consist of 0s and 1s with a 1 (resp. 0) at (i,j) if yj does (resp. not) appear in xi (according to the translate of subscripts on (*7). Then, applying Ix,y to the transpose of  [y1 … yn] (so it is a column vector) gives the column vector of the xis. Denote the transpose of Ix,y by trIx,y. From the difference set definition, notice:

trIx,y ×  Ix,y  = Ix,y ×  trIx,y = k-1 In + u1n×n, with In the n×n identity matrix, and 1n×n the matrix having 1s everywhere.

Apply both sides to the transpose of  [y1 … yn],  to conclude the matrix trIx,y has rows giving the difference set attached to inverting the relation between the xs and ys. Now look at the last column of Ix,y. A 1 appears at position j if and only if row 1 has a 1 at column n-j+1. That is, mod n, column n is -1 times row 1 translated by 1. That concludes the last line of DS3.

On numerology alone, we may consider which triples (n,k,u) from DS3  afford difference sets. These are the only possibilities up to n=31:

(*8) (7,3,1), (11,5,2), (13,4,1), (15,7,3), (16,6,2), (19,9,4), (21,5,1), (22,7,2), (23,11,5), (25,9, 3), (27,13,6), (29,8,2), (31,6,1).

The cases n = 22 and 23 and 27 are eliminated by the Chowla-Ryser Thm. which I discovered in [Ha, Thms. 3, 4 and 5]. It says, for n even (resp. odd), existence of a difference set implies k-u is a square (resp. z2=(k-u)x2+(-1)(n-1)/2y2 has a nontrivial integer solution. Hall's book suggests Chowla-Ryser is if and only if for existence of a difference set. Still, we now know for sure, if there were such a converse, it would not produce a different set in a doubly transitive design, because we now know the Collineation Conjecture is true (§V.4).

The next section hints at which groups – and conjugacy classes – arose as monodromy of Davenport pairs. This appearance of projective linear groups, combined with Riemann-Hurwitz, shows why we stopped the list of (*8) with n=31. This was the first inkling of the Genus 0 Problem.

V.3. Misera's example (sic): Take a finite field Fq (with q=pt for some value of t, p a prime). For an integer v ≥ 2, consider Fqv+1 as a vector space V over Fq of dimension v+1. Then, the projective linear group PGLv+1(Fq)=GLv+1(Fq)/(Fq)* acts on the lines minus the origin in (Fq)v+1: on the points of projective v-space, Pv(Fq). Take n=(qv+1-1)(q-1). Conclude:  PGLv+1(Fq) has two (inequivalent) doubly transitive permutation representations, on lines and on hyperplanes.

Further, those representations are equivalent as group representations by an incidence matrix that conjugates one representation to the other. Finally, here is the crucial point of what Misera told me. Consider a cyclic generator, γq, of the nonzero elements of Fqv+1. Such exists by the generalization of Euler's Theorem. Then,  γq acts by multiplication on Fqv+1  (identified with (Fq)v+1). It induces (as does $(\gamma_q)^{q-1}$) an $n$-cycle in $\PGL_{v+1}(\bF_q)$ acting on PGLv+1(Fq).

From Misera's example, I had confidence I was meeting upon a nontrivial, albeit subtle, phenomenon, in DS2. This was near the end of my first year at IAS. The most serious step was this: 

DS4: [Fr73, p. 134] writes difference sets for n=7=1+2+22, 11, 13=1+3+32, 15=1+2+22+23, 21=1+4+42 and 31=1+5+52. My notes to Feit show there are Davenport pairs (f,g) (§ II.2) over some number field for each of these cases.

In rereading [Fr73], I see [Fr73, (1.25)] left out n=15 in its list of difference sets. I'll do that case now for use below.

Take an irreducible degree 4 polynomial over Z/2 (say, x4+x+1). Then, multiply the nonzero elements (nonzero linear combinations of 1, x, x2, x3 corresponding to 1, 2, 3, 4) by x and use the relation x4+x+1=0, to label them 1, 2, …, 15. Example: x4 = x+1 corresponds to 5.

Choose a hyperplane: Say, the linear combinations of 1, x and x2. Then, a difference set (D15={1, 2, 3, 5, 6, 9, 11} mod 15) arises by listing elements of this hyperplane. 

Def. of Multiplier: A multiplier of difference set D mod n, is c ∈ (Z/n)* with cD a translate of D mod n. Denote by MD the group of multipliers of D. 

Example: 2 is a multiplier of D15, generating MD15, an order four subgroup of the invertible integers mod 15. A translate of the one [CoCa99, §2.2.5] took is {1, 2, 3, 8, 10, 13, 14}. After multiplication by -1, this is a translation of D15.

Here as for n= 7, the nonmultipliers of the difference set consist of the coset of multipliers time -1, compatible with the contribution of Storer from the opening of §VI. In that section we refer to γq as σ∞. We do that here to allow directly refering to the following observation. Use the notation of § App.1, with q = pt.  A choice of σ∞, up to conjugacy, defines the inertia generator from §III.4 attached to a polynomial f  with geometric monodromy  between PGLn(Fq) and PΓLn(Fq). Further, σ∞, up to conjugacy, defines the attached difference set up to translation given in (*7).

Multiplier Lemma: Inside PΓLn(Fq), the subgroup of (Z/n)* that corresponds to powers of σ∞ conjugate to σ∞ equals MD.

V.4. Group theory immediately after Graduate School: Ax – with whom I went to Stony Brook (though I left after getting tenure), instead of to University of Chicago where I was first offered tenure – suggested that I should explain what I was after to Walter Feit. His rationale: While my difference set conditions were complicated, group theory could handle astoundingly intricate matters by comparison to what one could do with algebraic geometry. From the wisdom of Ax's suggestions, I  learned to partition a problem into its group theory, number theory and Riemann surface theory pieces, precisely enough that I could handle each separately.

My interactions with Feit were complicated – in those days all through regular mail. The idea of what I expected was this. The case n = 11 is special for it corresponds to a difference set with a doubly transitive group of automorphisms that doesn't fit into the points-hyperplane pairing on a projective space over a finite field. Still, my reading suggested that I now knew all possibilities for these doubly transitive designs – exactly as described in §VI.3 – through Riemann's Existence Theorem. Here was the group theory guess.

Collineation Conjecture: A group with two inequivalent doubly transitive permutations representations, that were equivalent as group representations (of degree n) and containing an n-cycle, must have either degree 11, or lie between PGLv+1(Fq) and PΓLv+1(Fq), n=(qv+1-1)(q-1), for some v and q.

Given the Collineation Conjecture, I could give branch cycle descriptions for all Davenport pairs, thus solving problem D2. This was based on knowing that each branch cycle moved at least half the points. I suggested this to Feit in my description of its consequences, and he proved it ([Fe70, Thm. 3] or [Fr73, Prop. 1]). I assumed the Collineation Conjecture, and described from it the only possible – finite set of – values n (as in the rest of this report) that gave Davenport pairs over some number field. Indeed, it gave the full nature of those pairs, as in § App.5, the hardest issue to explain to algebraists. I also now knew none from these had definition field over Q (§ VI.2).

So, then by applying Riemann-Hurwitz this cut down the total number of branch cycles, in general. Yet, without the Collineation Conjecture, it did not make a case for the Genus 0 Problem. 

Still, Feit suggested that if I accepted the simple group classification, then extant literature might prove the Collineation Conjecture. I took that advice, allowing me to finish the Collineation Conjecture (its publication appeared much later in [Fr99, §9]), and several other pieces of pure group theory. §VII.3 gives an example that models how a (non-group theorist) researcher might approach using modern group theory.

Yet, the biggest surprise didn't come from group theory. It was possible to finish Davenport's Problem over Q without the Collineation Conjecture – or anything related to the classification of simple groups – using a device whose general  applicability opened up directions that went far beyond discussions of separated variables. The next section explains this, and relates my only specific mathematical interaction with UM beyond graduate school (see §VII.4).

VI. The B(ranch)C(ycle)L(emma) and Solving Davenport's Problem: 

I was immensely assured – at the time (§VI.2) by Storer's (the next) Statement. Notice, however, that the second sentence of DS3 – which I overlooked at the time, but made use of later – already gives its main thrust,  By assumption Tf and Tg are distinct permutation representations. If, however, -1 was a multiplier, then they would not be. That doesn't take away from the assurance I got from his statement, and its effect on the eventual impact of the BCL.

Storer's Statement: [Fr73, p. 132] says this: "According to T. Storer the fact that -1 is not a multiplier is an old chestnut in the theory of difference sets. He has provided us with a simple proof of this fact, upon which we base the proof of Lemma 5."

What I learned – in the examples of §V.3 – was that no Davenport pair has polynomial pairs over Q. This was thanks to the BCL, a general result on covers and  maybe the easiest general tool for divining properties of algebraic relations. That finished Davenport's Problem over Q with no additional group theory, and certainly no use of the classification.

VI.1.  The action of GQ: We denote the group of automorphisms of the algebraic numbers by GQ. If all elements of GQ  fix an algebraic number, then that number is rational (in Q).

Algebraic relations have coefficients. If the coefficients are algebraic numbers (lying in some finite extension K, of Q), then, points with algebraic number coordinates satisfying these relations determine all points satisfying the algebraic relations. This is called Hilbert's Nullstellensatz. 

§ II.1 reminds of the distinction between affine sets (defined by equations in a finite set of variables) and projective sets (defined by homogeneous equations in a finite set of variables). In practice this means that if you take any algebraic set and act on an algebraic point of it by γ∈GQ, then the image point will lie on the set defined by the γ acting on the old equations' coefficients.

Suppose we have a degree n rational function f in x. Then, points of P1z that are the image of fewer than n points of P1x under f are branch points, z1,…,zr, of f.  To be explicit with our polynomial covers, we'll take zr to be ∞. What we now say works for any (ramified) cover of P1z, not just a polynomial cover.

If the coefficients of  f are fixed by γ∈GQ, then γ  permutes z1,…,zr. Each branch point zi corresponds to a conjugacy class Ci in Gf (as in §III.4). Denote by ei the order of elements in Ci, and by N = Nf the least common multiple of the eis. We need to distinguish between the conjugacy class Ci in the geometric monodromy group, and the class aCi (defined by any element in Ci) in the (possibly) larger arithmetic monodromy group aGf (§II.3). chow-coh-zass-conjs.html (or its attached paper [Fr95]) gives many examples in the service of (An, Sn)-realizations of this distinction.)

The B(ranch)C(ycle)L(emma) says the permutation of the branch points corresponds to a permutation of the aCis put to an integer power cγ;∈(Z/N)*): the integer given by γ: e2π i/N → ecγ2π i/N. That is, if γ maps zi to zj, then

(*9)   aCj = aCicγ (all elements of aCi put to the power cγ ∈(Z/N)*).

BCL Consequence: For f with coefficients in Q, the cover γf of P1z from applying γ to f, is exactly the same as f. So, the new conjugacy classes must – in some order – equal the old. In particular, if zi ∈Q, then aCi = aCic, for each c ∈( Z/N)*. That is, aCi is a rational conjugacy class.

Recall that a regular (field) extension L/Q(z) is one for which the only constants in L consist of Q. So, the BCL applies to deciding possibilities with what conjugacy classes you can achieve a group as a regular Galois extension of Q(z). Also, it can predict the distinction between aGf and Gf. It does this by checking for any group H in Sn that contains and normalizes Gf, whether the BCL Consequence  holds by using H in place of aGf for some permutation of {1,2, …, r}.

That is, by considering H in the paragraph above, the BCL can often identify configurations of branch points that could allow a cover to have Q as definition field while a larger field is the definition field of its Galois closure. The problems of Davenport, Schur and Serre (the latter his Open Image Theorem) are sensitive to this.

VI.2. Applying the BCL to Davenport's Problem: If a polynomial f has coefficients in K, then the total ramification over ∞ (regard this as a  K point) implies, with n = deg(f), that a geometric component of the Galois closure has a subfield of K(e2πi/n) as its definition field ˆKf (§I.3). 

Let γxi (resp.  γyi)  be solutions of the equations γf(x)-z=0 (resp. γg(y)-z=0)) from applying γ to the respective coefficients of f and g. For each c∈(Z/n)*, choose any γ∈GK whose restriction to Q(e2πi/n) is c. This gives an action of (Z/n)* on the equation (*7), producing a relation
γx1=γyc+ γycα2+…+γycαk.
By taking these solutions to be Puiseux expansions at ∞, you confidently trace this action. Take Df = {1, α2,…, αk} to be the corresponding difference set (as in DS3).

DS5: Suppose (f, g) is a Davenport pair – with f indecomposable – over some number field K: the hypotheses of D2 (or DS2, but over K). Then: K contains the fixed field Q(Df) of the multiplier Mf of Df in Q(e2πi/n). More generally the following conclusions hold.

1. Since -1 is not a multiplier (Storer's Statement),  Q(Df) is not contained in the reals. So, for any Davenport pair, K is not Q, thereby solving  #3 of § II.2 with  the hypothesis that f is indecomposable. 
2. For each of the degrees n in DS4, there exist Davenport pairs over K if and only if it contains Q(Df). For n=7, 13, 15, and only in these cases, there are infinitely many Davenport pairs modulo Möbius equivalence (as in §I.1).
3. For the degrees given in #2, there are in fact, covers f with branch points defined over fields disjoint from Q(e2πi/n). For those, and for τ ∈ GQ  acting trivially on the branch points, but sending e2πi/n to e-2πi/n. These Davenport pairs (f,g) satisfy f(x)=τg(x) (action on the coefficients by τ).
        
That #1 is really about conjugacy classes follows from the Multiplier Lemma (§V.3). The multiplier group is measuring how far from rational is the conjugacy class of σ∞. Also, #1 follows from the deduction in DS3 that -1 times the difference set defined by f in a Davenport pair (f, g) gives the difference set for g. Since g and f give  inequivalent covers, this implies the difference set for multiplication by -1 could not have been a translate. I didn't, however, make that observation in [Fr3].
 
#2 (whose precise form gives #3) was the most serious mathematics.  Here is why I went after its general context, which takes up the rest of §VI. While Schur's Conjecture was easy compared to Davenport's problem, there were other problems, much tougher, that acceded to the method here, including Serre's Open Image Theorem and the Hilbert-Siegel problems. Clearly I think "attempting to write equations out" is not a road to success. Yet, many want to see equations, a topic that UMStoryCoeffics-Equats.html revisits.

Lewis knew Al Whiteman, who had been at the Institute my first year there. I had seen him talk on difference sets, his speciality. He gleaned that I was onto a problem that could use someone expert in difference sets and that I was returning to Michigan to talk on Schur's Conjecture. His student Storer had been hired by Michigan.

In fact, I stayed over a month during the summer of '68 to start writing up [Fr73]. Several times Storer and I discussed at his office blackboard. The combinatorial trick [Fr73, (1.19)] is Storer's. It was also valuable hearing from Storer, who often told his opinions of me, whenever it came to his mind. Especially:  There must be something wrong with me for knowing so much mathematics. His thought: It must be because I spent all of my time slaving in the library. (For the record: I learned mostly by being attentive at talks – not just those in my immediate areas; secondly from seriously refereeing hard papers. That's relevant to my comments on using group theory in §VII.3.)

Storer's opinions made an interesting compound with what I heard about me from Al Whiteman's wife thirty years later – at the funeral of Dennis Estes at USC (and long after Whiteman had died).  She noted what all the mathematicians's wifes remembered about me was one Summer night of '69 at a huge Stony Brook conference outdoor party, that I had shocked them all.

"With what," said I. Her response: "The way you danced!"

Yes, there was a real band, and a gorgous hippie-type passed by. She said she wasn't much of a dancer. Yet, I urged – must have been within earshot of others – "Just follow me," and she did. (A few days previous, the Lunar Excursion Module – on which I had worked as a full-time aerospace engineer during 3 years between the time I was 18 and 21 – had landed on the moon. I thought going to Stony Brook auspicious. Alas, it was an algebra-hating department.)

VI.3. Producing Davenport pairs: Here are rhetorical questions whose answers give a precise form to Statement #2 of DS5. Use notation from §VI.1. 
  1. What accessible data would allow easily concluding there is at least one Davenport pair (f,g) over some number field with f indecomposable having one of the degrees 7, 11, 13, 15, 21 and 31?
  2. Given an affirmative answer to #1, what structure might you find for all such Davenport pairs, and what definition fields for each of these degrees?
  3. What has this to do with simple groups, and what tools might you consider to dissuade others from searching for Davenport pairs of degrees other than those in #1?
  4. Assuming success in the above, what general conclusions might you dare about monodromy groups of polynomials or rational functions?
VI.4. Branch cycles, the tie to group theory: Recall σ∞ in §III.5, a generator of inertia over ∞. In §VI.1, whatever the branch points, z1,…,zr, for some cover given by a function f : X → P1z on a compact Riemann surface, these produce representatives, σ1,…, σr, of conjugacy classes C={C1,…,Cr} by the same process – a finger walking (again, I choose clockwise) around the branch point zi of the cover, producing a closed path Pi. The element  σi is the permutation of the points over the base point by following that path. For a polynomial cover, always assume zr is ∞.

The index, ind(σ), of a permutation σ in Sn is just n minus the number of disjoint cycles in the permutation. Example: an n-cycle in Sn has index n-1, and an involution has index equal to the number of disjoint 2-cycles in it. The Riemann-Hurwitz formula says the genus, gf of X satisfies

(*10)     2(n + gf + 1) = ∑i=1r ind(σi).

I will start with the case n=7, to show how the computations work, but then refer to the most interesting case (n=13) to compare with the work of others who have considered the production of equations. The group PGL3(Z/2) acts on 7 points and 7 lines of 2-dimensional projective space over Z/2. An involution (order 2 element) fixes the points on a line (of 3-points); every other nonidentity element fixes no fewer points. 

§VI.5 shows why there are Davenport pairs with their geometric monodromy group equal to PGL3(Z/2). This will answer question #1 of §VI.3, for degree 7. The method works for all degrees in question #1.

First consider r=4, and consider what would be the minimal possible indices for branch cycles of a polynomial f  with monodromy group PGL3(Z/2), where σ4 is a 7-cycle.  Then, the minimal possible sum of the four indices of corresponding σis is 3.2+6=12. In our case the right side is 12, so the genus is 0, and no other choices with r=4 would produce genus 0.

Since the cover for a polynomial map has genus 0, this is apt. Further, if such a polynomial exists representing f in a Davenport pair, we now know that these σis, i=1, 2, 3, all lie in this hyperplane fixing conjugacy class. One difference set here is {1, 2, 4}. So, we can ask if there is a Davenport pair  (f,g) with the permutation representation for f acting on {1,2, …,7}, so that an inertia group generator over z=∞, acts as (1 2 … 7) in the permutation representation Tf, while it acts as translates of {1,2,4} for Tg. 

What we need is a converse – cover producing conditions – from such σis. There is one: R(iemann)'s E(xistence) T(heorem).  Given such σi, i=1,…, r, in a group G, we are asking when there is a cover f : X → P1z branched at any given points, z1,…,zr, with its geometric monodromy group G, and having the attached conjugacy classes C={C1,…,Cr} of  σ1,…, σr. The answer: There is one if and only, for some σi' conjugate (in G) to σi, i=1,…, r, these conditions hold.
  1. Generation:  <σi', i=1,…, r> = G ≤ Sn; and
  2. Product-one: σ1'… σr'=1 (so, in #1, any r-1 of these elements generate G).
Those who use the monodromy method call such σi's satisfying #1 and #2 branch cycles. We call the collection of all such, in the respective conjugacy classes C , the Nielsen class Ni(G, C) of the cover. Further, covers corresponding to two such choices of r-tuples satisfying #1 and #2 will be isomorphic as covers (of P1z) if and only if some element in Sn conjugates the one r-tuple to the other.

Computing a Nielsen class correctly requires knowing the group NSn(G,C): the subgroup of Sn that normalizes G, and permutes the conjugacy classes in  C (preserving their multiplicity).

§ App.3 explains why branch cycles give covers, and why those covers are algebraic. (In the Davenport cases, each represents a polynomial map.) Observe: The genus gf in (*10) depends only on the images of σ1,…, σr in Sn, corresponding to the permutation representation Tf. For that, the conjugacy classes C don't seem important.

Still, Davenport's problem already exposes (as in the Multiplier Lemma) that conjugacy classes of  n-cycles are significant in the projective linear groups. As in DS5, say using Storer's Statement, there is more than one such class. In §VI.5 this tells us why the covers we produce – though they give Davenport pairs – are not over $\bQ$, but over a larger number field.

For n=7 there are two, represented by σ∞ and σ∞-1. For n=13, {1,2,4,10} (translation equivalent to {0,1,3,9}) is a difference set [Fr05, p. 60], with multiplication by 3 a multiplier: σ∞a, with a running over the powers of  3 mod 13.  So there are 4 (translation) inequivalent difference sets mod 13.

VI.5.  Computing and Using a Nielsen class: Continue with n=7 and the conjugacy classes of the previous section. Condition #1 of §VI.4 is necessary (with G=PGL3(Z/2)) to assure we get the pair of doubly transitive representations. It is easy to write, by hand, all involutions that could appear as σ1, σ2 or σ3. Example: If in (*7), use as a hyperplane that containing the fixed points corresponding to 1, 2 and 4.

Then the involution is one of (3 5)(6 7), (3 6)(5 7) or (3 7)(5 6). Conjugate by (powers of) σ∞ to get all the rest. To find possible 3-tuples (σ1, σ2, σ3) with product the specific 7-cycle σ∞-1=(7 6 5 4 3 2 1), is now simple (done in detail in [Fr95, p. 349]). Therefore the covers with fixed branch points ( z1, z2, z3, ∞), and fixed conjugacy classes attached to these in a given order) correspond to this absolute Nielsen class:
Ni(PGL3(Z/2), C)ab = Ni(PGL3(Z/2), C)/PGL3(Z/2). 
By listing the 4th entry as σ∞-1, the only further normalization to fix absolute Nielsen class elements is given by conjugation by σ∞. There are precisely seven such.  Such a 4-tuple (σ1, σ2, σ3, σ∞), produces f.

Apply Tf to the four entries to get (σ1', σ2', σ3', σ∞'), the branch cycle description for g: the other half of its Davenport pair. The most significant part of the monodromy method is that it often can make precise statements about the collection of covers in a given Nielsen class. Here is an example of that that means. 

DS6: Denote Q((-7)½) by Qf,g. Infinitely many (Möbius inequivalent – §I.1) Davenport pairs exist over any extension K of Qf,g. They correspond to the values of a parameter t7 in K. There is a similar result, and corresponding cyclotomic field Qf,g and parameter tn, for n=13 and 15.

§ App.5 shows the braid computation that gives DS6, thereby dispelling any mystery about Qf,g. The parameter t7 appears because there is just one component of Davenport pairs. Under reduced equivalence of the Davenport pairs, the parameter space is a curve, and the genus of that curve is 0. The strong properties of the parameter for these cases results from the transitivity of a braid subgroup on the Nielsen class. Generally, however, there is always a natural curve and the procedure shows how to list the components of the curve, and calculate the genus of these components.

Is it my imagination, or do I hear you asking something like, "So, where these Davenport pairs?" Well, § App.5 also gives references for their specifics, including later, alternate treatments – based on this one – that actually produce these pairs.

VII. The significance of Davenport's Problem:

In looking over what came from Davenport's Problem, and the other problems accomplished by the monodromy method, there remain two  general problems – one about Chow motives, and the other about going beyond the simple group classification – that will require new techniques. We explain those problems and then return to the comment from [So01] (see § I.3) on the groups that occur 'in nature' being close to simple groups.

VII.1. The Genus 0 Problem: The lessons I learned from Davenport's problem were lucky. How could I have guessed they would come about. Most propitious was my interaction with John Thompson, walking to lunch one day early in Fall 1986 after I arrived at the University of Florida. I gave him my conviction of the specialness of genus 0 monodromy groups related to polynomials. My support came much from [Fr80].
  1. The product-one condition (#2 of §VI.4)  together with genus 0 were the source of limitation on the groups arising in Davenport's Problem, and the Hilbert-Siegel problem (as in [Fr74a]). Each problem seemed to generate a limited set of genus 0 groups outside the list in #1, though each also had further constraints on conjugacy classes, and the major group constraint of primitivity.
  2. Using just polynomials, the geometric monodromy groups Sn, An, cyclic and dihedral appeared often. These examples distinguished covers over Q versus their Galois closures being defined over larger fields. 
Comments on #1: My main question to John was whether he thought that genus 0, product-one and primitivity would be sufficient to limit exceptional groups, and what exactly for this problem exceptional would be.

Comments on #2: Geometric An/Arithmetic Sn polynomial covers ((An,Sn)-realizations) gave the counterexamples to the problems listed in [Fr95]. Tchebychev polynomials have dihedral geometric monodromy and their Galois closures are defined over the maximal real field in Q(e2πi/n). Serre's OIT even more dramatically showed that rational functions with dihedral monodromy gave dramatic distinctions between arithmetic and geometric monodromy. In the Hilbert-Siegel problems (first discussed in [Fr74a]), the unique exceptional case, polynomials of degree 5 with geometric monodromy A5, produced another representation on unordered pairs of integers, that gave a relevant rational function. That was a hint that the permutation representation on pairs (ordered and unordered) of integers for An or Sn could appear often as a geometric genus 0 monodromy.

Thompson's response – immediately he confessed to being "seized" by the problem – was that I shouldn't limit it to polynomial covers. Rather, include  indecomposable rational functions (genus 0 covers). In place, however, of considering constraints and guessing what precisely the exceptional permutation representations might be, he suggested just to show that all composition factors of the geometric monodromy groups would be cyclic or alternating. Then, the exceptions would come from just finitely many simple groups (outside An and, of course, cyclic). In that case, the reduction to primitive was automatic and natural.

All statements related to Schur problems, and especially the interpretation of dihedral groups (whose composition factors are cyclic) as the essence of Serre's OIT suggested we should aim at a statement on actual monodromy groups rather than composition factors. Still, what John proposed was a start that would generate data.

He proposed we work on that together. My heart was in algebraic equations. I suggested Bob Guralnick as far more appropriate. Here was the upshot recounted in Genus0-Prob.html with more precise mathematical detail.


Peter Müller produced a definitive classification of the polynomial monodromy, including – a la what happened in Davenport's Problem– a list of the polynomial monodromy that arose over Q [Mü95]. Davenport's Problem had captured the harder "exceptional cases" of that classification.

The more optimistic conjecture I made for polynomials turned out true even for indecomposable rational functions. That is, it was possible to consider the precise permutation representations that arose in series of groups related to alternating and dihedral groups. This addition to Guralnick-Thompson was Guralnick's work (and formulation) with many co-authors and independent papers by others.

Guralnick visited Florida while I was there, and he and Thompson generated series of groups based on running through the classification of primitive groups using [A-O-S] (§ VII.3 and § App.2). [A-O-S] constructs a template of five patterns of primitive groups. Into four of those you insert almost simple groups. The fifth was comprised from affine groups. This then naturally divided the task into running through the simple groups inserted into these templates, a special expertise of Guralnick (see § VII.3). 

So, the Genus 0 Problem ran through two filters: [A-O-S]; and the distinct series of finite simple groups. This enumeration accounts for the number and length of the papers contributing to the problem's resolution. This rigamarole is what gave Guralnick the data to so precisely formulate the final result.

I could look at early Guralnick-Thompson results on exceptional genus 0 groups from this list, and just from the BCL see a small number provided rational functions outside Serre's OIT that gave Schur covers over Q: one-one maps on Z/p ∪ ∞, for infinitely many p. We didn't know such existed previously. Further, by being so precise, in each case, it was possible to give qualitative statements for all genuses, not just genus 0.

That is, by precisely distinguishing difficult cases that happened to have genus 'slightly' larger than 0, the distinctions between genus 0, 1 and higher genus came clear. Qualitatively, however, it was not possible to be so precise on most of the exceptional "genus groups" (even for genus 0) when it came to such series as the 'exceptional simple groups of Lie-type.'  Davenport's Problem and the other problems that arose early in these developments remain the unequaled archetype for being precise.

VII.2. Attaching a zeta function to a diophantine problem: Schur's Conjecture and Davenport's Problem have simple statements using Chow motives (which have attached zeta functions).

For Davenport, the statement interprets to a zeta function being trivial. It was Ax's idea to consider attaching a naive zeta function to any similar Diophantine problem. Yet, there was no way to compute it or find its properties, until [Fr76] introduced Galois Stratifications (annals76.html has a brief review). This was my replacement for Chow motives, which didn't then exist.

Denef and Loeser later showed how to make this zeta attachment canonical, using Chow motives. Still, their proof went through Galois Stratification. So, Davenport's Problem was my foray into mathematical objects studied by others that were in the kernel of the linearization of Diophantine problems using Chow motives. My conclusion: This kernel is often what much practical mathematics is about. ChowMotives.html defines Chow motives and uses Davenport's Problem as an example.

VII.3. Applied group theory and todays challenges occuring 'in nature:'  The topic of what groups occur 'in nature' started in § I.3 with a phrase of Solomon [So01]. If the snippet I've used as a surrogate were 'in Nature,' then I might use [LS08]  (authors based at UM) in the Scientific American as a substitute. Their article found a way to sneak in the topic of 'what are simple groups?' Still, this spirited analog of Rubik's cube was based on not much more than – like S8 – a Mathieu group M12 property: it is generated by two elements. (By the way, as a consequence of the classification, so are all the other simple groups.) 

One serious goal of [So01] was to document that the simple group classification – including the so-called quasi-thin part questioned by Serre [Se92, 94] – is available. That is, mathematicians may apply it in the various ways suggested here, with confidence. Yet, the monodromy method – as used in Davenport's Problem – will require most mathematicians to get some aid from a group theorist.

To show how cooperating with group theorists would work, I later took on one more problem in the Davenport range. That was a version of Schur's problem restricted to finite fields of a fixed characteristic, but still just about polynomials. Guralnick and Jan Saxl joined me on in the 3rd section: Going through every step of the [A-O-S] classification as in § VII.1 and § App.2. Though we didn't complete the affine group case, the results were definitive. We knew all exceptional polynomials whose groups either had cyclic inertia over $\infty$ or where not affine, except for the primes 2 and 3. Even for the primes 2 and 3, we had restricted the possible monodromy groups – these were $\PSL_2$ groups – and degrees, and through a chain of papers, those groups and degrees did give new exceptional polynomials. Further, we solved an 1897 conjecture of Dixon.

I was not a passive purveyor of Guralnick and Saxl. First, I caught the unusual new Schur covers for the primes 2 and 3 that were slipping by overly-optimistic group assumptions. Second, I carefully showed how using [A-O-S]  worked (§ App.2). The original proof of Schur's conjecture applied, easily, to describe all exceptional (Schur) polynomial covers of degree prime to the characteristic. When this hypothesis did not hold, other than when the degree equals the characteristic – Dixon's 1897 conjecture – the inertia group is likely no longer generated by a single element σ∞, ( from § III.3).

The replacement is that the group G has a factorization: It is a set theoretic product of the stabilizer of a letter in the permutation representation, and the (small, but not cyclic) inertia group at ∞.  Further, without primitivity we would have been lost. So, we could only assume that  aGf was primitive, and Gf maybe not. What I understood was that organizing [A-O-S] was Guralnick's job, while filling in possible factorizations of primitive groups that could arise was Saxl's – based on his familiarity with [LPS].

McLauglin might have approved. It resembled how he often laid out the steps that allowed him his seemingly-encyclopedic recall in our two-person seminar at UM in Fall 1966. Yes, the monodromy method works.

Being based on RET, the classification, Chebotarev analogs, etc., yes, they probably are a lot to handle, though one might notice I was a kid – with only two undergrad years (in electrical engineering), and three years of grad school – when I did. Still, I return to the phrase from [So01] to note that along with serious progress, there remain serious challenges.

For examples, there are challenges to any insistence that it is only (or mainly) groups close to simple groups that occur 'in nature.' I stick here with the challenge of non-primitive groups (leaving to my web site how Modular Towers is a very different challenge) as it arises in extending Davenport's problem.

Indeed, I can say the challenges with just the next step of the Davenport-Schinzel problems. If f is indecomposable, statement DS2 says the two problems are equivalent. I conclude this section by showing each problem poses its own diverse challenges once you drop the indecomposable assumption. First consider Davenport's Problem (over Q).

Peter Müller has gone after finding exceptions from polynomials with exactly two composition factors. His list [Mü98, p. 25] considers f(x) = a(b(x)) ∈ K[x] with K a number field. He assumes b does not form a Davenport pair over K to another polynomial b*. Otherwise, you can compose both b  and b* with any a and get an obvious  Davenport pair. His conclusion: then g has the form a(b*(x)). He lists the small number of groups that arise for this situation.  Still, he notes [Mü98, p. 27] an old acquaintance from DS2: Tf and Tg are equivalent as group representations in all examples that appear to date. Finally, he leaves a conjecture for the case K = Q.

Müller's Conjecture [Mü98, Conj. 11.3]: Let f, g ∈ Q[x] be a Davenport pair over Q. Then, they are either linearly equivalent or f(x) = h(x8) and g = h(ay8) for some polynomial h ∈ Q[x] and constant a. (Davenport had already noted that (x8, ay8) form a Davenport pair when y8-a has a zero mod p for almost all p, say a=16. Of course, f and g are linearly equivalent, just not over Q.)

An example of the distinction between Davenport's problem and Schinzels starts on [Fr87, p. 17] under the heading of the (m,n)-problem. What that ask is if, for a 'general' pair (f',g') of polynomials (over the complexes), of respective degrees m and n, is it true that

(*11) no matter what are the nonconstant polynomials f''(x) and g''(y),  f'(f''(x))-g'(g''(y)) is irreducible.

[Fr87, p. 18] has branch cycles for such (f'(f''(x)),g'(g''(y))) of degree 4, given any (f',g'), both degree 2, so that f'(f''(x))-g'(g''(y)) reducible. That is, the (2,2) problem is false. Note, however, from the branch cycles, that (f'(f''(x)),g'(g''(y))) is not – over any number field – a Davenport pair.

Actually, it suffices to take for (f',g') any polynomials of respective degrees m and n giving simple-branched covers, and (outside ∞) disjoint branch points. Then, the (m,n) problem holds if, for some nonconstant f''(x) and g''(y) (their degrees are irrelevant), f'(f''(x))-g'(g''(y)) is irreducible.

Let N be the least common multiple of m and n. Then, the reduction in DS1shows it suffices to consider deg(f'')=kN/m, deg(g'')=kN/n.

For example, in the (2,3)-problem:  it suffices to consider f''(x) and g''(y) of respective degrees 3k and 2k. [Fr87, Prop. 2.10] shows neither k=1 or 2 gives a contradiction to (*11). Still, there was a close – group theoretic – call already with k=2 based on dealing with imprimitive representations.

Solomon didn't define the phrase 'appearing in nature' and maybe he won't consider these problems as being 'in nature.' I'm prepared for that because I have experience – if given a chance – to pose many problems that probe aspects of group theory from the disciplines of his list.

VII.4. Final UM Comments: In case it isn't clear, I think I learned much, and took great advantage, from my three years at UM. I was (almost) never frightened by prestigious mathematicians, or by being on my own in hot-house mathematical environments. What was the biggest problem was support for recognition. Even papers solving long unsolved problems appearing in prestigious journals didn't do much for either myself or those who found those problems attractive.

My career has (barely) survived by my interactions with European and Israeli mathematicians, doing a lot of what they wanted me to do, rather than what my own convictions suggested. When late in my career, I turned to the topics I'd put aside for many years, again I found that support for publication disappeared.

There were over 200 grad students at UM with me. I have seen only one student from my graduate years more than once after grad school. That was the topologist Bob Edwards who twice sat in on talks of mine at AMS conferences. It would have helped if other UM students, even slightly related, interacted with me from the hundreds of talks I've given, and the many conferences I've attended or run.

The three others who got PhDs in 1967 were all analysts, one much more famous than anyone who might be reading this. That was "The Unabomber," a no-show at the going away party Paul Halmos gave us. You can find a picture of me from years related here –  opposite the page with Grothendieck – in Halmos' "I have a photographic Memory." I was standing in front of my Schur Conjecture diagram at the end of my 1968 UM lecture on it.

I didn't know about that picture until many years later, just prior to my giving a talk at a conference packed with arithmetic people whose affiliation was Harvard, one way or the other. Several at that conference, held at University of Arizona at Tempe, were visibly upset that I had maneuvered to give an hour talk there. This was thanks to Armand Brumer being a no-show. He conceded his spot to me.

I discovered Halmos' picture by accident during the coffee break before my talk, while I was purposely off in a side commons room. It was appropriate inspiration – showing a 25 year-old me, facing the UM audience, in a confident pose – to help me handle with equanamity giving a talk to a likely antagonistic audience friedHalmos-Book.pdf (friedHalmos-Book.html thanks the AMS and says more about the picture).

Brown and Kister (from whom I had a course in vector/micro bundles) had left that earlier UM talk during a discussion session at which Davenport held forth on my talk. They came up to me later, to tell me why they left. They were annoyed by Davenport's remarks, which seemed to suggest that there was nothing new in what I had done. Halmos's picture had a surprisingly sympathetic caption under it about the mathematical direction I seemed to be going, perhaps influenced by how well I had handled Davenport's "interrogation."

Halmos' picture helped me do better than just get through that Tempe Arizona talk. Still, either I, or the Schur Conjecture, must have been funny. Once I saw that picture, I realized it was the answer to a New Yorker cartoon I had puzzled over years before. That cartoon – posted on my Stony Brook colleague's (Paul Kumpel) office door –  featured (I now saw) me charicatured for being satsified with that Schur Conjecture diagram.

Appendix on Group Theory and Branch Cycles

The relation between primitive groups and simple groups starts by recognizing that the two most common sets of highly non-abelian groups are symmetric groups and general linear groups, and both are in evident ways close to simple: Sn (close to An) ; and GLn(Fq) (close to PSLn(Fq)). Here q is a power pt of some prime p. We call these groups almost simple for those values n≥ 5 (resp. n and q, excluding n=2 and p=2 or 3) for which An (resp. PSLn(Fq)) is simple [Ar, Thm. 4.10].

§ App.1. Some useful group definitions:

The goals of algebraic covers and group theory don't match perfectly. In the latter's 20th century haste for the simple group classification it often could strip a group to an essential core, tossing essential data for covers. We give the full definition of almost simple, to show what it means to get to that core. Still, by staying with primitive groups – a concept natural for covers – § App.2 reminds of a tool sufficient, modulo considerable expertise, for handling covers from knowledge of simple groups. According to [GLS] a quasisimple group G is a perfect central cover G → S of a simple group S. Here: cover means onto homomorphism; perfect means the commutators g1g2g1-1g2-1 in G generate G; and central means the kernel is in the center of G. Such a cover is a special case of a Frattini central cover where the map, if restricted to a proper subgroup of G, won't be a cover, but we don't assume S is simple.  Then, if S is perfect, so is G.

A component, H ≤ G, of G, is a quasisimple subgroup which has, between H and G, a composition series – a sequence of groups each normal in the next.  The group generated by components and the maximal normal nilpotent subgroup of G is called the generalized Fitting subgroup, F*(G), of G.  [GLS] calls a group G almost simple if F*(G) is quasisimple.

Notice that we don't lose the quasisimple property if we extend PGLn(Fq) to PΓLn(Fq) by adjoining a Frobenius, Frp (pth power map on coordinates),  for Fp to PGLn(Fq). It extends the permutations on lines and hyperplanes; actually, on linear spaces of any dimension. The notation differs slightly from what is used today, but [Ca, Chap. XII] with its many exercises, is where I learned about these groups in graduate school.

Supersolvable means that G has a chief series – a series of subgroups normal in G (versus just normal in the next in the series), each of prime index in the next [Is, p. 133].

An affine group is a subgroup of the full group of actions of GLn(Fq) and translations on the vector space (Fq)n of dimension n over Fq. The case that arose in Burnside's Theorem (§ IV.1) is n=1.

§ App.2. How could I possibly have understood the group theory in [FrGS]?:

Technically, this section is how [FrGS] uses [AsS]. Still, informally it is an outline of how I – with only informal training in group theory – could have been the main writer of the paper, not passing muster on the Part III, until I understood it.

The problem here is exactly the Schur Problem #2 of §II.2, except it is over a given finite field Fq. The hypothesis is the one-one mapping property for a polynomial f for infinitely many extensions, Fqt, of Fq. [FrGS, Part II] establishes a list of group properties of the Galois closure of f. These allow a characterization using the A(schbacher)-O('Nan)-S(cott) Classification (Theorem [AsS]) of primitive groups. Excluding affine groups, there are four other primitive group types, whose form has the shape of dropping almost simple groups into particular positions. Five points about this process call for clarification.
  1. Reduction to the case aGf is primitive (in its natural permutation representation; so aGf(1) is a maximal proper subgroup of aGf) is almost trivial. Also, if the degree of f  is prime to q=pt (or it is p itself) the technique of the original proof of the the Schur Conjecture worked easily.
  2. Unlike the §II.2 version of Schur's Conjecture, if the degree of f is not prime to p, you can't easily reduce #1 to the case Gf (the geometric group) is primitive.
  3. Though the ramification group I∞ over ∞ is no longer generated by a single element, σ∞ (§ III.5), a loosening of this statement works: There is a set theoretic factorization aGf(1). I∞ of aGf.
  4. [FrGS, Part III] starts by clarifying the definitions in [AsS]. Then, it combines it with a classification of the appropriate factorizations of the groups that arose from [LPS]. The result excluded all but easily classified exceptional covers, and those with Gf either a. an affine group (and degree of f a power of the characteristic), or b. one of the two groups PSL2(Fpa) with p=2 or 3, in the respective characteristics 2 or 3, with a odd.
  5. It was possible for me to follow and coordinate the respective analyses of my two co-authors, especially assuring the PSL2 cases didn't slip through.
An addendum to #1 is that it was a Dickson Conjecture from 1896 that the degree p case was as we described it. Very quickly, upon publication of [FrGS], Cohen and Matthews, Müller, and  Lenstra and Zieve produced examples of the PSL2 exceptional polynomials for p=2 or 3 and all odd a as in #4. With that addition to #4, [FrGS] had given a rigamarole for selecting Galois properties that synched nicely with the classification, to nail all possible solutions to the original problem, with one exception – the case of affine groups.

Since my whole point here is about the collaboration between algebraic geometry and group theory, the most important addendum is to #5. I could no more have completed this result alone, without much more confidence in the classification than I had the right to, than a certain Nobel physicist (Alvarez) could have ascertained that iridium fell in one stratigraphic layer near the K-T boundary in the Hell's Creek formation in Northeast Montana. This required collaboration with a topnotch paleontologist (Clemens). Nor has academia found a formula for apportioning the significance and interpretation of such respective contributions, unless, of course some authors were put in fakely. This was not so in either of the two cases I've now mentioned. Finally, what got attention of others was the unanticipated surprises in the results, for which there is only one word: Luck, albeit collaborative skill, and care, exposed them!

A statement due to Wan, that an exceptional polynomial should have degree prime to q-1, was immediate from [FrGS] before Wan formulated his conjecture. It wouldn't have occurred to the authors of [FrGS] to take that conjecture seriously, until we found that others mistakenly took its statement to mean that elementary methods had achieved our result. Wan's statement told so little about exceptional polynomials, not even their degrees, while [FrGS] characterized so much. Even in the one mystery, the precise monodromy group in the affine case in #4 –, it has the degree of f a power of the characteristic [carlitz-quick.html].

§ App.3. Branch cycles produce an algebraic cover:

It is the existence of special – classical – generators (classicalgens.pdf) of the fundamental group of P1z\ {z1,…,zr}=Uz that we get the major unsolved problem in the use of RET. Given a set of classical generators, there is an explicit one-one correspondence between branch cycles and algebraic covers of the sphere branchcd over {z1,…,zr} in a Nielsen class.

Classical generation problem: Both sides of this correspondence have an algebraic description, but classical generators are not algebraic. The problem is to prove such a correspondence without using them or some other such topological gadget. Alg-Equations.html discusses special cases of this problem to show why easy cases, say accessible from Kummer theory, don't give a hint at the difficulty for, say, covers with almost simple monodromy, not even polynomial covers arising from Davenport's Problem.

[Mu76, p. 27] discusses the relation between Teichmuller and Torelli space by listing an equivalent to classical generators, but this is brief, and imprecise.  While the applications in [Vo] seem to be only about the Inverse Galois Problem, that is misleading. [Vo] is suitable for such applications of RET as those we discuss here – for the obvious reason that it is these that motivated developments behind [Vo]. A slower self-contained treatment, filling in everything from material in Ahlfors book is in [Fr09, Chap. 4]. Right to the heart of the matter is the much briefer Nielsen-Classes.html.

It is not immediate that having a cover f: X → P1z means that X is algebraic. Still, that follows easily from Chow's Lemma (§ VI.1) once you have a single further function that separates – has different values on – the fiber over any point of Uz. It is the R(iemann)-R(och) Theorem that in general guarantees such a function. No one argues over that theorem, though it is certainy non-trivial.

When X has genus 0, shouldn't it especially simple to produce such a function (lets call it w). But is it? Here is an historical track to finding w. You take the differential df of f. From general principles it has degree 2gX - 2 = -2. Similarly, for the function w (once we have it). It's differential dw has degree -2.

An especially good w would be one that separates all points (is an isomorphism of X to P1w). The support of its polar divisor is concentrated over w=∞. Since X is simply connected, any meromorphic differential with this property, being locally integrable, is globally integrable to a function. Uniformizer Problem: When gX=0, what types of data allow automatic creation of such w giving the isomorphism X to P1w?
§ App.4. Grabbing a cover by its branch points and braids:

§VI.4 points to the essential object – a set of classical generators on the r-punctured sphere Uz' – that assigns an absolute Nielsen class element to a cover. Suppose you start with one cover branched over z', and then deform the punctures z', to another set of r distinct points z''. Then an automatic analytic continuation of the cover follows the path of the branch points (Hurwitz-Spaces.html, §V).

The Hurwitz monodromy group: Now consider the case z'=z'', an equality of unordered branch points. That is, a closed path L: t ∈ [0,1] → z'(t) in Ur (continuous and piecewise differential). Such a a path might actually permute the order of the points in z'. Along that path we also can deform the initial classical generators P', so as to end up with a new set of classical generators P'' at the end of the path.

A base point distinct from the branch points is necessary to talk about classical generators. Therefore, freely following L may force us to deform the base point z0', too: t ∈ [0,1] → z0(t)', with z0'=z0(0)'$ and z0''=z0(1)'.

If z0'' does not equal z0', you can add a further deformation leaving z' fixed, just for the purpose of getting the original base point. Mapping the beginning classical generators P' to the end classical generators P'' induces an automorphism of π1(Uz',z0'). There is no canonical way to deform z0'' back to z0'. So, to make this automorphism unambiguous requires modding out by the conjugation action of π1(Uz',z0') on itself.

Running over all such paths L induces the Hurwitz Monodromy group, Hr, acting as automorphisms of π1(Uz',z0') modulo this inner action. Two elements of Hr generate it. We call these q1 and sh. For our purposes we have only to know their action (given in [Fr77, §4]) on a Nielsen class representative: g=(g1,g2,g3,…,gr)
Conjugating q1 by sh, gives q2, the 2nd twist, the twisting action moved to the right. Repeating that conjugation gives q3, etc. A permutation representation of a fundamental group – in this case, of Ur – produces a(n unramified) cover H → Ur. Each point p of H       represents an equivalence class of covers. The equivalence here is the simplest possible (called absolute): equivalence f: X → P1z and f'': X' → P1z if there is a continuous map from X to X' that commutes with the projections to P1z.

Denote the space of r ordered points on P1z by Ur. Points of the pullback, H ord of H → Ur to Ur represent covers f together with an ordering on the branch points of a cover.

Statement on Hurwitz space points: Let K be a (characteristic 0 field). Then, each K point of H (resp (resp. of H      sym) corresponds to an equivalence class of covers defined over K (resp. with its branch points also defined over K). When a cover f has no non-trivial automorphisms commuting with this projection, there is a unique total family of covers over H           /;.  (or over H      sym). Such a total family is represented by a (ramified cover) T → H× . P1z. In that case, a K point p of H gives a well-defined K cover T p → p× . P1z, which we interpret as a K cover of P1z. There is a similar statement for H      sym [Fr77, §5] or

Several problems call for only considering paths that keep a point fixed, which we take as zr=∞. Foremost among these are problems considering spaces of polynomial covers. Suppose a cover f: X → P1z has X of genus 0, and a totally ramified place over zr=∞. Then, in the isomorphism of X with P1w, with no loss assume w'=∞ maps to zr=∞. Then, there is a polynomial P : P1w → P1z giving a cover equivalent to f.

For any r ≥ 4, put an ordering on the branch points, and regard r-2 of the finite branch points, z2,…, zr-1, ∞, as parameters. Then, consider the locus H      orz1 of H      or in which only the first finite branch point z1 varies. This is an algebraic curve whose nonsingular compactification, H      orz2,…, zr-1, ∞, is a ramified cover of the locus over points P1z1×(z2,…, zr-1,∞), where only z1varies.

§App.5 uses Davenport polynomials of degree n=7 to show how to compute the genus of components of this curve and another natural, related curve that appears when r=4.  From computations on branch cycles, we know over which fields there are Davenport pairs of the allowable degrees (#1 of § VI.3).

§ App.5. Three genus 0 families of Davenport Pairs:

In our problem we have a Hurwitz space of covers P1z, and in the cases n=7, 13 and 15,  r=4. There are two algebraic curves that arise in consider those Davenport pairs that have r=4 branch point covers for their general case of realization. Example: Let D denote the difference set {1,2,4} mod 7 in example n=7 of §VI.5. Then, [Fr95, p. 349] lists representatives of the seven elements – here denoted 1', 2', …, 7' – in the the absolute Nielsen class Ni(PGL3(Z/2), C)=Ni of §VI.4.

The number of components of the Hurwitz space is the number of orbits of <q1,q2,q3> on Ni.  Expression [Fr95, (4.14)] calculates these elements and by inspection you see there is just one orbit, so just one component.

Then, [Fr95, (4.15)] calculates the action of

h1 = q1-2, h2 = q1q2-2q1-1, and  h3 = q1q2q3-2q2-1q1-1 on Ni.

For example, h1 is (1' 3' 5')(2' 7')(4' 6'). The product of these elements is a relation in the Hurwitz monodromy group H4, and so they represent branch cycles for the cover H      orz2,z3, ∞ → P1z1×(z2, z3,∞), as in  §VII.4. Again by inspection you see there is one orbit of <h1, h2, h3> Ni,  and by R-H applied to this case (instead of  (*10)), the genus of H      orz2,z3, ∞ is gD = 0, computed from 2(7 - gD -1)=3.4. 

Since the degree of the cover is odd (7), that automatically implies the function field of H      or, is Q(-7½, a,b,tD) where a,b, and tD  are algebraically independent indeterminates, representing (respectively) the branch points z2, z3 and a generator of the genus 0 function field of H      orz2,z3, ∞. No parameter here represents ∞ because we consider polynomial pairs, and we won't let ∞ move to other values. Using the Statement on Hurwitz space points in App. 5,  conclude that over every finite extension of Q(-7½) there are infinitely many (significantly different) Davenport pairs, even having branch points defined over that field.

[Fr99, Thm. 8.1 and 8.2] shows the case n=13 work similarly, and as easily. Here the Hurwitz space is a degree 13 – again the same as n – cover of U4. The only significant difference is that since the multiplier of the difference set D={1, 2, 4, 10} in this case has order 3, the definition field K for these spaces is the degree 4 extension of Q inside Q(e2πi/13). Therefore, there are two pairs of conjugate Davenport pairs in this case.

[Fr05, §3.4]

Assume:

(*11) P is not a functional composition P''oP' with P' a nontrivial cyclic polynomial.

Then, there are no automorphisms that commute with the projection P. Each point p ∈H                      corresponds to an equivalence class [Pp] where the collection consists of the set Pp(a'x+b'). The full Möbius class of polynomials includes these polynomials composed with the transformations of P1z that send P to   aP(x)+b. The resulting space of equivalence classes is called the reduced space Hrd. I've published the (early)  examples (§V.3) around Davenport's problem as elementary – yet, significant – families of covers in later papers.
RETURN polynomial covers D7 of P1z. Davenport pairs of degree 7 is a degree 7 unramified cover of




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