Algebraic Equations and Finite Simple Groups:
What I learned from graduate school at University of Michigan, 1964–1967

PRELUDE: After three years of Graduate School in Mathematics at University of Michigan (1964-1967), writing a thesis under the direction of Don Lewis, I left for a postdoctoral at the Institute for Advanced Study. There I studied with Goro Shimura. My first year was extended to two years (1967-1969). Then, I went with James Ax to SUNY at Stony Brook. After receiving tenure and a Sloan Fellowship, I left. That bare bones outline of a beginning career tells little mathematically.

It has no hint that the work inspired by my time at UM connected resolutely with the simple group classification – through conversations with John Thompson – and with modular curves – interactions with J.P. Serre. Nor, that problems of Andrzej Schinzel and Harold Davenport, (visitors to UM my second year) in papers with Lewis, were the inspiration. Not even that technical tools came from assiduous use of Grothendieck's fiber product approach to algebraic equations. Yet, fulfilling those connections required – no word better – tutoring from many UM-affiliated faculty during my formative years.

This short version of the story relates Davenport's Problem, the steps in its solution, and how the connections above came about. A fuller, referenced version is at UMStory. That shows I never lost my youthful enthusiasm for completing programs of Abel, Galois and Riemann, as recorded in "What Gauss told Riemann about Abel's Theorem."

DAVENPORT'S PROBLEM AND FIBER PRODUCTS: When number theorists say almost all primes p, they mean all but finitely many. Davenport sought relations between two polynomials f(x) and g(y) with rational coefficients – where no change of the variable in f gives g – having the same ranges on integers mod p for almost all p. He liked this style of question, and often used exponential sums to interpret it.

Changing "almost all" to "infinitely many" and taking g(y)=y, restates the hypothesis of Schur's 1921 conjecture. The conclusion of Schur: f must be composed of linear, cycle and Chebychev polynomials. Richard Brauer was a student of Schur, and the advisor of Don Lewis. When I met him (see below) he asked if I knew he had worked on Schur's conjecture. I hadn't.

A variables separated algebraic equation looks like f(x)-g(y)=0. Writing this as f(x)-z=0 and g(y)-z=0 opened the territory. Fiber products of f and g over the z-line allowed me to use groups to draw conclusions. I'll use n for the degree of f.  Visiting Assistant Professors Armand Brumer and Richard Bumby guided my mastering Grothendieck's Tohoku paper and pieces of his EGA. There I learned to go between algebraic equations and group theory.

Chuck MacCluer's thesis, under Lewis, showed – for special f  – a geometric statement gave Schur's one-one mapping hypothesis. Later, by extending the Chebotarev density theorem, I formulated a general context including Davenport and Schur. There, pure group theory translated the number theory.

Formulations, however, are not conclusions. Could you invert the direction polynomials to groups, as in Schur's Problem?

A rational function not composed of lower degree functions is indecomposable. I state lightly what I found powerful in practice.

1. A rational function's covering group is primitive exactly when the function is indecomposable.

2. A polynomial's covering group always contains an n-cycle.

From that I learned, that if f was indecomposable, its covering group was either doubly transitive, or f was in the Schur conclusion. That finished the Schur story and deepened the Davenport story.

I now knew Davenport's hypothesis on f and g produced a difference set mod n that encoded how zeros of g(y)-z=0 summed to a zero of f(x)-z. Not only did the distinct permutation representations for f and g have the same degree, their group representations were identical.

I had seen Brumer pepper Jack McLaughlin with group theory questions. When Brumer left for Columbia at the start of my 3rd year, I took his place with McLaughlin from whom I learned the distinction between doubly transitive and primitive. Richard Misera, a fellow grad student working with Donald Higman, saw this interaction and gave me a propitious example, coming from projective linear groups. I applied this, modulo something that I learned very much on my own – R(iemann)'s E(xistence) T(heorem) – to produce polynomial pairs having almost simple groups with special projective linear core. The three propitious points were these:

1. Without writing equations, I was able to see the Galois action of the cyclotomic field of n-th roots of 1, acted on the difference set relating f and g. The elements that preserved that difference set, up to translation (so-called multipliers), gave the definition field of the pair (f,g). Further, -1 was never a multiplier, so that definition field was never Q.

2. Because the covers given by f and g had genus 0, the only possible degrees for f and g were n=7, 11, 13, 15, 21, 31.

3. The cases with infinitely many essential pairs (f,g) modulo mobius action on z, x and y appearing in #2, had degree 7, 13 and 15. Further, in these cases those essential parameters formed a genus 0, upper half-plane quotient, that wasn't a modular curve.

Tom Storer, newly at UM, when I visited it from IAS, worked with me on the last statement of #1. This completed Davenport's Problem over Q for indecomposable polynomials f. There were no nontrivial examples. It used nothing from the simple group classification. The offshoot of that technique became the Branch Cycle Lemma, far and away the most practical tool by which to relate  geometric covering groups and definition fields.

USING THE CLASSIFICATION AND THE GENUS 0 PROBLEM: I conjectured that the only Davenport pairs possible over any number field were those I'd listed above. Ax started me consulting with Walter Feit, who suggested – modulo the simple group classification – a paper by Kantor would give that. It did. The idea is this.

A Theorem of A(schbacher)-O('Nan)-S(cott) classifies primitive groups as arising from constructions with either (almost) simple groups at their core, or the group is affine. Adding that the group came with an n-cycle allowed me to complete the result from Kantor's paper on my own.

William Leveque, with whom I had written a "first thesis" in diophantine approximation, had translated a famous paper of Carl Ludwig Siegel into English. His notes opened my understanding of theta functions, started me using techniques of Riemann, and showed me how to interpret Abel's theorem as an arithmetic statement, a la Andre Weil's thesis. That foray into Siegel's powerful works, and a course on the fundamental group taught by Morton Brown, enabled me to read, and immediately use beyond its contents, Springer's book on Riemann Surfaces. From these I had translation techniques extending the Chebotarev theorem, to apply to many problems.

The main trick – even with Siegel and Weil – was recognizing many unsolved problems as about variables separated equations. To these my methods on Davenport's Problem applied immediately. Here is the hardest, my favorite.

A Hilbert-Siegel Problem: Let f(x) be a polynomial with Q coefficients. Suppose the set of integers z' for which f(x)-z' is reducible, but has no degree 1 factor, is infinite. This happens automatically if f decomposes, but the only others have degree 5 (and then there are examples).

The group translation was to what I called a primitive group with a double-degree representation: That is a group with a doubly transitive representation and an n-cycle, which had another representation of twice that degree.

A positive came out of this: Monodromy groups of indecomposable polynomials were very special, and likely classifiable. Excluding finitely many exceptional covering groups, they came from well-known permutation representations of cyclic, dihedral, alternating and symmetric groups. The degree 5 special case in the Hilbert-Siegel Problem showed me how actions by the symmetric group on sets of  integer pairs might occur, but even for the alternating and symmetric group, polynomial monodromy was limited.

Walking to lunch with John Thompson one day at University of Florida, I gave him my conviction, listing my data above.

His 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). Then, a likely statement would have all composition factors cyclic or alternating. He proposed we work on that together.

My heart was in algebraic equations. I suggested Bob Guralnick far more appropriate for that problem. Here was the upshot.

Peter Mueller 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. Davenport's Problem had already captured the harder elements of that classification.

The more optimistic conjecture I made for polynomials turned out true even for indecomposable rational functions. This addition to Guralnick-Thompson was Guralnick's work with many co-authors. Here, however, it was not possible to be so precise on most of the exceptional "genus 0 groups" in the manner of the exceptional degrees that came out in Davenport's Problem.

UM SEMINARS AND MODULAR CURVES: Double transitivity of a permutation representation is an easily understood property. Primitivity much less so. I immediately, relayed to Lewis' number theory seminar my early discoveries, like #1 and #2 above, and this:

3. If f is a polynomial, then it is indecomposable over the definition field exactly when it is indecomposable over the algebraic closure.

Serre's Open Image Theorem is a statement on absolute Galois groups acting on systems of modular curve points. It has two parts: Called, respectively, complex multiplication and GL₂. I began my work on moduli space problems by translating the Schur Problem for rational functions to Serre's Theorem. The GL₂ case showed myriad rational functions violate #3. Later Guralnick-Mueller-Saxl showed there were only sporadic other examples. This, too, used the Davenport Problem method.

Bob MacRae was a visiting UM assistant professor who attended the Number Theory seminar. He wrote up two of my thesis sections for publication. These related #1-#3 to other general properties of variables separated equations. These papers are far more quoted than my two (later) Annals papers. MacRae also arranged for me to talk at U. of Chicago through his advisor Irving Kaplansky. Later Richard Brauer, who attended that talk, presented me with a tenure offer during my 2nd post-doctoral year at IAS. Those two sentences are still full of mysteries to me.

I saw how the major Diophantine results on modular curves interpret as a statement on regular realizations of Dihedral groups as Galois groups with the covering ramification given by involutions.

Later I generalized this to any p-perfect group and conjugacy classes prime to p a direction backwards to Davenport's Problem. It produced a (Modular) Tower of spaces – rarely of modular curves, but often very modular curve-like – to interpret the problem. No rational points at high levels has now been shown in many cases, a result from various incarnations for modular curves by Demjanenko-Manin, Faltings, Mazur, then Mazur-Merel. Modular Towers completely encodes many unknown aspects of the Inverse Galois Problem: My version of Shimura Varieties.

Lewis arranged that I could go to Bowdoin college NSF-funded summers of eight weeks each on Algebraic Number Theory (1966) and Algebraic Geometry (1967). Both summers I learned everything put in front of me. I also learned that I would be regarded as an ignoramus for not having the background prevalent at Harvard, MIT or Princeton at the time.

Notes of Brumer, following Gunning, and a seminar at UM with Roger Lyndon and I as alternating speakers, prepared me for many aspects of modular curves. I was ready for Princeton, and for getting the most of my interactions with Shimura.

THE SIGNIFICANCE OF DAVENPORT'S PROBLEM: The Schur Conjecture and Davenport's Problem have simple statements using Chow motives (which have attached zeta functions). For Davenport, the statement interpret to a zeta function being trivial.

It was Ax's idea to consider attaching a zeta function to any similar Diophantine problem. Yet, there was no way to compute it or find its properties, until my 1976 Annal's paper introduced Galois Stratifications. This was my replacement for Chow motives, which didn't exist then. 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.

Further, much practical mathematics on equations has their variables separated. (Think hyperelliptic curves and slight generalizations to see how prevalent that is.) The resolution of the genus 0 problem is an apt tool to figure where exactly variables separated fits among all two variable algebraic relations.

Of course, without an aid to help with group theory, you can't use the method. I later took on one more problem in the Davenport range, for which I needed help from group theorists. That was a version of Schur's problem over a single finite field. Guralnick and Jan Saxl joined me on in the 3rd section: Going through every step of the A-O-S classification. Though we didn't complete the affine group case, the results were definitive. That included solving an 1897 conjecture of Dixon.

I was not a passive purveyor of Guralnick and Saxl. First, I caught the unusual examples of new Schur covers that were slipping by overly-optimistic group assumptions. Second, I carefully showed how using A-O-S worked. 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.

FINAL UM COMMENTS: This story supports those who believe in the essential connectedness of mathematics. My fuller telling (UMStory) shows the achievements had influence on the work of many. Yet, official success is another story. Taking a geometric approach to number theory doesn't bode well when a totally number-oriented approach– as in the Fermat's Last Theorem/modular curve connection – has gleaned the most attention.

Further, making disparate connections in an era of technical specialists cuts the possibilities of readers sharply. Finally, when I look, say, at a Scientific American, or a New York Review of Books, it comes clear that these magazines are so aware they must avoid serious discussion of mathematics, despite how many areas of science build on it.

John Thompson once asked me what Davenport thought of my methods. My response: "He didn't like group theory, or Galois Theory, or even algebraic geometry. I got the impression he hated them." John nodded: "That sounds like him."

Still, it was Lewis' role as the algebraist that brought both Davenport and Schinzel into my world. Many people have said they first heard of me from Shinzel's 1970 International congress talk, that alluded to his problems solved by the monodromy method above.

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 and run. There were over 200 grad students at UM with me.

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 years later. A brief description of the picture is at fried-HalmosBook.html. Still, either I, or the Schur Conjecture, must have been funny. A New Yorker magazine not long afterwards based a cartoon on it.

I have seen only one person 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.