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The Mathematical Contributions of Serge Lang, Volume 54, Number 4 The Mathematical Contributions of Serge Lang Jay Jorgenson and Steven G. Krantz his article is the second in a two-part editions and revisions? For example, he wrote two series in memory of Serge Lang, who books, entitled Cyclotomic Fields I and Cyclotomic passed away on September 12, 2005. In Fields II, which were later revised and published the first article, which appeared in the in a single volume, yet his text Algebra: Revised May 2006 issue of the Notices ,weinvited Third Edition has grown to more than 900 pages Tcontributions from a number of individuals who and is vastly different from the original version. knew Serge on a somewhat personal level. For this In an attempt to determine how many books he part, we sought expositions which would describe, wrote, we consulted the bibliography from Lang’s with certain technical details as necessary, aspects Collected Papers, where he highlighted the entries of Serge’scontribution to mathematicalresearch. which he considered to be a book or lecture note. To begin understanding the breadth and depth According to that list, Lang wrote an astonish- of Serge’s research endeavors, we refer to Volume I ing number of books and lecture notes—namely 1 of his Collected Papers, where he outlined his sixty—as of 1999. Furthermore, he published mathematicalcareerina numberofperiods.We list several items after 1999, there are more books here Lang’s own description of his research, using in production at this time, and a few unfinished only a slight paraphrasing of what he wrote (see manuscripts exist. In 1999, when Lang received tableattopofnextpage).Forthisarticle,theeditors the Leroy P. Steele Prize for Mathematical Ex- choosetousethislistasaguide,thoughitshouldbe position, the citation stated that “perhaps no obvious that we cannot address all facets of Lang’s other author has done as much for mathematical mathematical research. exposition at the graduate and research levels, In addition to research, Lang’s contribution both through timely expositions of developing to mathematics includes, as we all know, a large research topics ... and through texts with an number of books. How many books did Lang write? excellent selection of topics.” We will leave it We (the editors) are not sure how to answer that to others to assess the impact of Serge Lang’s question. Should we count political monographs books on the education of mathematics students such as Challenges? How do we count multiple and mathematicians throughout the world; this Jay Jorgenson is a professor of mathematics at City Col- topic seems to be a point of discussion properly lege of New York and the Graduate Center. His email addressed by historians as well as by history address is [email protected]. itself. Steven G. Krantz is deputy director of the Amer- On February 17, 2006, a memorial event was ican Institute of Mathematics. His email address is held at Yale University in honor of Serge Lang. At [email protected]. that time, Anthony and Cynthia Petrello, friends Based on mathematical contributions from: Alexandru of Serge since the early 1970s, announced their in- Buium, Jay Jorgenson, Minhyong Kim, David E. Rohrlich, tention to create a fund for the purpose of financ- John Tate, Paul Vojta, and Michel Waldschmidt. ing mathematical activities in memory of Lang. As 1Serge Lang: Collected Papers. Vols. I–V, Springer-Verlag, mathematicians, we the editors express our sin- New York, 2000. cere thanks to Anthony and Cynthia Petrello for 476 Notices of the AMS Volume 54, Number 4 1. 1951–1954 Thesis on quasi-algebraic closure and related matters 2. 1954–1962 Algebraicgeometryand abelian(or group) varieties; geometric class field theory 3. 1963–1975 Transcendental numbers and Diophantine approximation on algebraic groups 4. 1970 First paper on analytic number theory 5. 1975 SL2(R) 6. 1972–1977 Frobenius distributions 7. 1973–1981 Modular curves, modular units 8. 1974, 1982–1991 Diophantine geometry, complex hyperbolic spaces, and Nevanlinna theory 9. 1985, 1988 Riemann-Roch and Arakelov theory 10. 1992–2000+ Analytic number theory and connections with spectral analysis, heat kernel, differential geometry, Lie groups, and symmetric spaces Chronological description of Serge Lang’s mathematics career. their generous support of mathematical research. closed constant field is trivial was achieved by As teachers, we see the story of Serge, Anthony, showing that such a field is C1, and he called that and Cynthia as a wonderful example of the type property quasi-algebraic closure. In analogy with of lifelong friendship that can develop between Tsen’s theorem, he conjectured that the field of instructors and students. As editors of the two all roots of unity is C1. This is still an open ques- articles on Lang, we are in awe at being shown tion. In his thesis Lang proved various properties yet another way in which Serge has influenced the of Ci fields and showed that a function field in people he encountered. We look forward to see- j variables over a Ci field is Ci j . He also proved + ing the results from the development of the “Lang that the maximal unramified extension of a local Fund” and its impact on the mathematical com- field with perfect residue field is C1. Artin had al- munity. so conjectured that a local field with finite residue field is C2. Lang could prove this for power series fields, but not for p-adic fields. In 1966 it became Serge Lang’s Early Years clear why he failed. G. Terjanian produced an example of a quartic form in 18 variables over the John Tate field Q2 of 2-adic numbers with no zero. Terjani- an found his example a few months after giving These remarks are taken from my talk at the Lang a talk in the Bourbaki Seminar (November 1965) Memorial at Yale on Lang and his work in the ear- on a remarkable theorem of Ax and Kochen. Call ly years, roughly 1950–1960. Lang’s papers from a field Ci (d) if the defining property of Ci above this period fill less than one half of the first of holds for forms of degree d. Using ultrafilters to the five volumes of his collected works. His pro- relate p Qp to p Fp((t)), they showed that for ductivity was remarkably constant for more than each primeQ p, QQp has property C2(d) for all but fifty years, but my interaction with him was most- a finite set of d. In that sense, Artin was almost ly early on. We were together at Princeton as grad- right. uate students and postdocs from 1947 to 1953 Lang got his Ph.D. in 1951. After that he was and in Paris during 1957–58. a postdoc at Princeton for a year and then spent In the forward of his Collected Papers, Lang a year at the Institute for Advanced Study be- takes the opportunity to “express once more” his fore going to Chicago, where he was mentored appreciation for having been Emil Artin’s student, by Weil—if anyone could mentor Serge after his saying “I could not have had a better start in Ph.D. He and Weil wrote a joint paper generaliz- my mathematical life.” His Ph.D. thesis was on ing Weil’s theorem on the number of points on a quasi-algebraic closure and its generalizations. curve over a finite field. They show that the num- He called a field k a Ci field if for every integer ber N of rational points on a projective variety in d > 0, every homogeneous polynomial of degree Pn of dimension r and degree d defined over a d in more than di variables with coefficients in k finite field satisfies has a nontrivial zero in k. A field is C0 if and only r r 1 r 1 if it is algebraically closed. Artin’s realization that N q (d 1)(d 2)q − 2 Aq − , Tsen’s proof (1933) that the Brauer group of a | − | ≤ − − + function field in one variable over an algebraically where q is the number of elements in the finite field and A is a constant depending only on n, d, John Tate is a professor of mathematics at the Universi- and r. (Here and in the following, “variety defined ty of Texas at Austin. His email address is tate@math. over k” means essentially the same thing as a ge- utexas.edu. ometrically irreducible k-variety.) From this result April 2007 Notices of the AMS 477 they derive corollaries for arbitrary abstract vari- B A of commutative group varieties, via a eties. They show, for example, that a variety over map→ α : V A defined outside a divisor. These → a finite field has a rational zero-cycle of degree 1. coverings, which he calls “de type (α, A)” can be The paper is a very small step towards Weil’s con- highly ramified, and Lang notes that in the case jectures on the number of points of varieties over where V is a curve, taking Rosenlicht’s general- finite fields which were proved by Deligne. ized Jacobians for A and throwing in constant Lang proved for abelian varieties in 1955 and field extensions, one gets all abelian coverings, so soon after for arbitrary group varieties that over that his theory recovers the classical class field a finite field k a homogeneous space for such a va- theory over global function fields. riety has a k-rational point. A consequence is that These papers were the beginning of higher- for a variety V over k, a “canonical map” α : V dimensional class field theory and earned Lang Alb(V) of V into its Albanese variety can be de-→ a Cole Prize in 1959.
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