Journal of Number Theory 133 (2013) 825–829

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David R. Hayes: Some remarks on his life and work

I am sure that I must have met David Hayes for the first time at two summer conferences thatwere held at Bowdoin College in the mid 1960s. One was about algebraic number theory and theother about algebraic geometry, subjects of great interest to both of us. However, try as I might,I cannot recall a specific interaction during that distant period.

I do remember quite vividly reading a paper of his entitled “Explicit class field theory for rationalfunction fields” [10] which appeared in the Transactions of the American Mathematical Society in1974. In this beautiful work, David exposits and carries forward some important work of his thesisadvisor Leonard Carlitz. Carlitz wrote over 770 papers (which may be a record). The one which Davidwrote about had the uninformative title “A class of polynomials”. In this paper Carlitz introducesa mathematical object which would later be renamed the Carlitz module. He showed that torsionpoints on the Carlitz module generate abelian extensions of F(T ), the rational function field over afinite field. These extensions are analogous to cyclotomic extensions of the rational numbers Q. Inaddition to giving an elegant exposition of Carlitz’s theory, David added the very important theoremthat if one adjoins all the torsion points the resulting field is “almost” the maximal abelian extensionof F(T ). Moreover, he shows how to modify the construction to give the whole maximal abelianextension, thus creating a function field version of the famous Kronecker–Weber theorem. In thesame year 1974, Vladimir Drinfeld published a paper (in Russian) entitled “Elliptic Modules”. In futureyears elliptic modules were called, for obvious reasons, Drinfeld modules. As it turned out, the Carlitzmodule is a very special case of a Drinfeld module.

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Hayes pointed out that already in his lectures Carlitz had constructed what would later be calledDrinfeld modules of all ranks, thus anticipating later developments many years before they occurred!See Hayes’ paper [17]. This paper is also interesting for the insight it offers into David’s experience asa graduate student at Duke.

To return to Drinfeld, the elliptic modules he defined enabled him to explicitly construct the max-imal abelian extension of an arbitrary global function field. A new and more elementary proof ofDrinfeld’s theorem was given by Hayes in a paper entitled “Explicit class field theory in global func-tion fields” which appeared in 1979 [11]. This was a marvelous accomplishment. For algebraic numberfields K one has an explicit constructions for the maximal abelian extension of K only in the casewhere K = Q or K is an imaginary quadratic extension of Q. It is amazing that in the function fieldcase, there is an explicit construction for an arbitrary base field.

These developments drew many new people into the area of arithmetic in function fields, myselfamong them. Soon I began running into David at conferences and corresponding with him on sub-jects of mutual interest. David developed rapidly into one of the most important contributors to thegrowing circle of mathematicians interested in this new approach to number theory in function fields.It continues to be a flourishing subject so that it is most appropriate to have this issue of the Journalof Number Theory dedicated to his memory and featuring new contributions to subjects in which hetook delight and to which he contributed so significantly. I will discuss in greater detail some of hislater research after giving a brief biographical sketch of his life.

David R. Hayes was born on July 14, 1937 in Raleigh, North Carolina to Woodrow (Woody) R. Hayesand Eleanor C. Hayes. His father was a musician who developed a sizable reputation in his fieldthroughout North Carolina. His mother was a homemaker. There were four children; David and histhree sisters Sally, Betsy, and Jenny.

David graduated from Needham B. Broughton High School in Raleigh in the spring of 1955. In 1959he graduated from Duke University with a B.Sci. degree in mathematics. He did his graduate work atDuke and wrote his PhD thesis under the direction of Leonard Carlitz. The title of his thesis is “Thedistribution of irreducibles in GF[q, x]”. He received his PhD degree in 1963.

David began his teaching career at the University of Tennessee in the fall of 1963 and continuedthere until the spring of 1966. In the academic year 1966–1967, he was a Postdoctoral Fellow atHarvard University. In the fall of 1967 he was appointed an Associate Professor at the University ofMassachusetts where he remained for the rest of his career, becoming a Full Professor in the fall of1972. He was a Visiting Professor at Oxford University in 1974–1975, at Harvard in the fall of 1981,at U. Cal. San Diego in the fall of 1983, at Imperial College of Science and Technology London in thespring of 1989, and at Harvard University in the fall of 1999. He was Chairman of the MathematicsDepartment at the University of Massachusetts from 1991–1994.

For many years David was a leading figure at the Five College Number Theory Seminar which isheld weekly at Amherst College. In the summer of 1991, David, along with David Goss and myself, or-ganized a large conference (over 100 participants from all over the world) at Ohio State University onthe subject of “The Arithmetic of Function Fields”. The conference proceedings were published a yearlater by Walter de Gruyter Publishers [7]. David Hayes and I also organized a three day conference “AMini-Conference on the Arithmetic of Function Fields” held at Brown University in April of 1996.

While on the faculty at the University of Massachusetts, David had six PhD students; Michael Nutt1974, Gove Effinger 1981, Mizan Khan 1990, Daniel Carter 1993, Zesen Chen 1996, and Maura Murray1999. David retired from teaching in 2002.

David’s first marriage ended in divorce. He had three sons by his first wife; Robert, Chris, and Jon.In 2004 he married Irene Brown. It was a happy marriage which was most unfortunately cut shortby his death in April of 2011. In addition to his sisters, his sons, and his wife, he leaves behind threegrandchildren, three step-sons, and two step-grandchildren.

Let us now turn from biography to a description of some of his many mathematical contributions.Being a student of Carlitz, it is natural to expect much of his work to concern the arithmetic theory ofglobal function fields. This is indeed the case, but there are also papers about number fields, compu-tational number theory, and elementary number theory. A theme of many of his works is the theoryof L-functions and the interplay between arithmetic and analysis.

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His PhD thesis and one of his first papers [8] concerns the distribution of irreducibles in arithmeticprogressions of the polynomial ring A = F[T ], where F is a finite field with q elements. Let a,b ∈ Aand suppose (a,b) = 1. Let Nd(a,b) be the number of monic irreducibles of the form ax + b whichare of degree d. In the 1920s, E. Artin proved the following about these numbers

Nd(a,b) = qd

Φ(a)d+ O

(qθd

d

).

Here, Φ(a) is a polynomial analogue of the Euler φ-function and θ is a fixed real number between0 and 1. Using the Riemann hypothesis for curves, one now knows θ can be taken to be 1/2. Davidproved a very general theorem of the same type when the irreducibles involved are subject to furtherrestrictions. For example, one may require the first s coefficients of such an irreducible to be specifiedin advance. Then, the above formula remains valid if we introduce a factor of qs into the denomina-tor of the main term. His thesis substantially generalizes (and corrects) earlier work by Carlitz andS. Uchiyama.

Most of David’s research concerns multiplicative number theory. However, one of his earliestprojects was to prove a polynomial version of a theorem of Vinogradov. This theorem states thatevery sufficiently large odd integer is the sum of three primes. In the ring A = F[T ] a polynomialf (T ) is said to be even if q = 2 and either T or T + 1 divides f (T ). Otherwise f is said to be odd.David was able to show that every odd polynomial of sufficiently large degree is a sum of three ir-reducible polynomials, see [9]. The proof is very original. Instead of using the proof of Vinogradov asa model, he uses an earlier “almost” proof of G.H. Hardy and J.E. Littlewood. This is based on theirfamous circle method and the GRH. The GRH for number fields is not known, even today. In functionfields the GRH is a theorem due to A. Weil. Thus, “all” David had to do was invent a circle methodfor the rational function field k = F(T ) and apply it successfully to his problem. He does invent a cir-cle method for function fields by using the adele classes Ak/k as his replacement for the unit circle.Applying the method involves a large number of delicate calculations, but all obstacles are overcome.

This is not the end of the story. Could one prove the result for all odd polynomials? Some restric-tions have to be made. The degree of our odd polynomials must be greater than one and, when q iseven, polynomials of the form x2 + a must be excluded. With these exceptions every odd polynomialf of degree d can be written as the sum of three monic irreducibles, one of degree d and the othertwo of smaller degrees, see [3] and [4]. This wonderful result was the joint effort of David and hisformer PhD student Gove Effinger. They first had to find ways to lower the bounds in the asymp-totic theorem to the point where a computer computation could check the remaining cases. Then along and very non-trivial computer calculations had to be done. According to [3] the computer cal-culation took 64.8 hours of supercomputer time on an IBM 3090. At the end, on December 19, 1989“A complete solution to the polynomial 3-primes problem was at hand”. A first rate result obtainedby ingenuity, originality, computational skill, and sheer persistence! The method and the details ofthe whole procedure are given in David’s only book (joint with G. Effinger) “Additive Number Theoryof Polynomials Over a Finite Field” [4].

Returning to the multiplicative theory, we noted earlier that the 1974 paper [10] which showedhow to generate the maximal abelian extension of F(T ) by adding torsion points on the Carlitz mod-ule was followed by a paper in 1979 [11] with the self descriptive title “Explicit class field theoryin global function fields”. Drinfeld had already shown how adding torsion points on certain Drin-feld modules enables one to generate the maximal abelian extension of an arbitrary global functionfield k/F. His proof was geometric and scheme-theoretic, very powerful, but difficult for many num-ber theorists to understand. David’s paper has two advantages. He puts the whole theory on a morearithmetic and elementary foundation thus making the beautiful theory available to a much larger au-dience. Secondly, he extends the result to a theory of ring class fields as well as ray class fields. Usingeither approach, the result is amazing. To generate the maximal abelian extension of Q one proceedsby adjoining all the roots of unity. If K is an imaginary quadratic number field one proceeds, roughlyspeaking, by adjoining the x-coordinates of torsion points on an associate elliptic curve. For no othernumber field is there an analytic construction of the maximal abelian extension. However, in the case

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of a global function field, adjoining torsion points on a certain carefully constructed rank one Drinfeldmodule works! More readable than David’s 1979 paper [11] is his elegant survey article “A brief in-troduction to Drinfeld modules” [16]. In this paper he exposits his theory of sign-normalized Drinfeldmodules and uses torsion points on such modules to construct ray class fields.

Once one has a cyclotomic style construction of abelian extensions of global function fields one canbegin to ask whether certain important theorems from the theory of cyclotomic number fields can becarried over to arbitrary global function fields. Steve Galovich and I made a beginning in that direction(see [5] and [6]) by working over the rational function field F(T ) and using the Carlitz module todefine analogues of cyclotomic units. We proved a Kummer style result relating class numbers to theindex of the group of cyclotomic units in the full unit group. David, in a series of deep and importantpapers, carried this theme way beyond anything we had thought possible. For general global functionfield he finds analytic class number formulas [12], elliptic units (ala G. Robert) and class number-unit index formulas in unramified abelian extensions [13], cyclotomic units, Stickelberger elementsand a proof of the Brumer–Stark Conjecture in global function fields [14], a proof of B. Gross’ p-adicrefinement of the abelian Stark conjectures in global function fields [15], and much else besides.

One of the main mathematical pre-occupations of David toward the end of his career were theStark conjectures, both in number fields and function fields. In 1999 while on leave at Harvard, Davidgave a series of lectures on the Stark conjectures. Notes for lectures 1–5 and 9 are still available onhis home page (which is still up as I write this). At one point he intended to extend these notes intoa monograph on the Stark conjectures. For some reason he gave up this project, which is regrettable.Given his lucid writing style and his deep insights into the subject, such a monograph would havebeen very valuable.

David’s interest and expertise in computation have already been mentioned in connection with hisjoint work with Gove Effinger. As further examples see his two papers with David Dummit [1] and [2].Early last year, when he was already very ill, his wife informed me that there were page proofs lyingaround the house. It turned out these were page proofs for a joint paper with Benji Fisher entitled“The 2-divisibility of h+

p ” which had been accepted for publication by the journal Mathematics ofComputation. I contacted Benji Fisher who then obtained the proofs, read them, and returned themto the journal. The paper should be appearing soon. It is undoubtedly David’s last publication.

I have mentioned only a portion of David’s research output, but more than enough to convey therichness of his interests and the significance of his contributions. I want to make some commentson David’s style as a mathematician. He was very serious about his work. He worked on subjectsof fundamental mathematical importance and he went deeply into them. He had excellent taste andan abundance of mathematical talent, but also resolve and persistence. If a seemingly insuperableproblem appeared, he would keep attempting a solution until all roadblocks were cleared and theway to further progress was open. I deeply admire his work. It is beautiful, important, and of lastingvalue.

Let me conclude with a few personal comments. As time went by our professional relationshipturned into a lasting personal friendship. We both liked the outdoors. During my visits to the Amherstarea we would often go for hike in the surrounding countryside. My wife and I have a vacationhome in New Hampshire. David would visit us there to hike in the autumn and ski in the winter.Of course, he and I would sneak in some mathematical discussions on the side, but mainly thesewere social occasions where everyone would enjoy the country air, the physical exertions, and thegood comradeship. We also enjoyed good food and good wine. We would spend at least part of theevenings indulging in both.

Many people who knew David describe him as very intelligent, quiet, modest, and gentle. This isall true. He was also very generous with his time and his ideas. In every way he was a person withexceptional qualities. It was a privilege to have known him and to have been his friend.

References

[1] D. Dummit, D.R. Hayes, Rank one Drinfeld modules on elliptic curves, Math. Comp. 62 (206) (1994) 875–883.[2] D. Dummit, D.R. Hayes, Checking the p-adic Stark conjecture when p is Archimedean, in: Algorithmic Number Theory,

Talence, 1996, in: Lecture Notes in Comput. Sci., vol. 1122, Springer, Berlin, 1996, pp. 91–97.

M. Rosen / Journal of Number Theory 133 (2013) 825–829 829

[3] G.W. Effinger, D.R. Hayes, A complete solution to the polynomial 3-primes problem, Bull. Amer. Math. Soc. 24 (2) (1991)363–369.

[4] G.W. Effinger, D.R. Hayes, Additive Number Theory of Polynomials Over a Finite Field, Oxford Math. Monogr./Oxford Sci.Publ., The Clarendon Press, Oxford University Press, New York, 1991.

[5] S. Galovich, M. Rosen, The class number of cyclotomic function fields, J. Number Theory 13 (1981) 363–375.[6] S. Galovich, M. Rosen, Units and class groups in cyclotomic function fields, J. Number Theory 14 (1982) 156–184.[7] D. Goss, D.R. Hayes, M. Rosen (Eds.), The Arithmetic of Function Fields, Walter de Gruyter Publishers, Berlin, New York,

1992.[8] D.R. Hayes, The distribution of irreducibles in GF[q, x], Trans. Amer. Math. Soc. 117 (1965) 101–127.[9] D.R. Hayes, An expression of a polynomial as a sum of three irreducibles, Acta Arith. XI (1966) 461–488.

[10] D.R. Hayes, Explicit class field theory for rational function fields, Trans. Amer. Math. Soc. 189 (1974) 77–91.[11] D.R. Hayes, Explicit class field theory in global function fields, in: G.C. Rota (Ed.), Global Function Fields, Studies in Algebra

and Number Theory, Academic Press, New York, 1979.[12] D.R. Hayes, Analytic class number formulas in function fields, Invent. Math. 65 (1) (1981/1982) 49–69.[13] D.R. Hayes, Elliptic units in function fields, in: Number Theory Related to Fermat’s Last Theorem, Cambridge, MA, 1981, in:

Progr. Math., vol. 26, Birkhäuser, Boston, Mass, 1982, pp. 321–340.[14] D.R. Hayes, Stickelberger elements in function fields, Compos. Math. 55 (2) (1985) 209–239.[15] D.R. Hayes, The refined p-adic abelian Stark conjecture in function fields, Invent. Math. 94 (3) (1988) 505–527.[16] D.R. Hayes, A brief introduction to Drinfeld modules, in: The Arithmetic of Function Fields, Columbus, OH, 1991, Walter de

Gruyter Publishers, Berlin, 1992, pp. 1–32.[17] D.R. Hayes, Introduction to Chapter 19 of “The arithmetic of polynomials”, Finite Fields Appl. 1 (2) (1995) 157–164.

Michael RosenBrown University, United States

E-mail addresses: mrosen@math.brown.edu, michael_rosen@brown.edu

29 July 2012Available online 15 September 2012

Communicated by David Goss