# Oeuvres — Collected Papers I: 1949-1959

###### Jean-Pierre Serre
Publisher:
Springer
Publication Date:
2013
Number of Pages:
596
Format:
Paperback
Series:
Springer Collected Works in Mathematics
Price:
79.99
ISBN:
9783642398155
Category:
Collection
[Reviewed by
Darren Glass
, on
02/22/2014
]

Every once in awhile I find myself in a conversation with someone about the greatest mathematicians of the twentieth century. Surely there are many good candidates for the title, but one name that I always come back to is Jean-Pierre Serre. There are many criteria one could use to rate the great mathematicians, and Serre rates highly in them all:

• One might measure the number of publications, and MathSciNet lists 319 publications by Serre as of this writing. Sure, this pales in comparison to the more than 1600 (and counting) publications that Erdős has, but it is nothing to sneeze at.
• One might look at the range of areas that they researched in. Serre started his mathematical life as a topologist — his thesis introduced a spectral sequence describing the homology of a fibred space in terms of the homology of the base and of the fibres — but he is probably best known for his work in algebraic geometry, although that description has to be construed broadly enough to include deep results in number theory, group theory, and representation theory. In a 1986 interview that appeared in Mathematical Intelligencer, Serre was quoted as saying “Some mathematicians have clear and far ranging ‘programs’ … I never had such a program, not even a small size one. I just work on things which happen to interest me at the moment.’
• One might compare the awards that mathematicians won. In 1954, Serre was the youngest person to ever win the Fields medal, and half a century later he was the first recipient of the Abel Prize. In between he won many other awards including the Wolf Prize, the Balzan Prize, and many honorary degrees and society memberships.
• In addition to being one of the great researchers, Serre is also an extraordinary expositor of mathematics. He wrote numerous books, including A Course In Arithmetic, Local Fields, Linear Representations of Finite Groups, and almost every description of one of his papers comments on the quality of the exposition. In 1995 he won the American Mathematical Society’s Leroy P. Steele Prize for mathematical exposition, and the citation for the award reads in part:
It is difficult to decide on a single work by a mathematician of Jean-Pierre Serre’s stature which is most deserving of the Steele Prize. Any one of Serre’s numerous other books might have served as the basis of this award. Each of his books is beautifully written, with a great deal of original material by the author, and everything smoothly polished. It would be hard to make any significant improvement on his expositions; many are the everyday standard references in their areas, both for working mathematicians and graduate students. Serre brings his whole mathematical personality to bear on the material of these books; they are alive with the breadth of real mathematics and are an example to all of how to write for effect, clarity, and impact.
• In addition to all of his writings on mathematics, Serre has also written on writing mathematics, and he has strong opinions on what makes a good paper. To this end, his (dare we say viral?) video lecture entitled "How To Write Mathematics Badly" is well worth watching.
• Another fact one could use to show the far-reaching influence of Serre is the existence of a Wikipedia page dedicated to Things Named After Jean-Pierre Serre, something that is not true of many other mathematicians.

If I have not yet convinced you that Serre’s name at least belongs in the discussion to be on a list of Greatest Mathematicians of the Twentieth Century, then I suggest you go out and read some of his works yourself, and you will quickly be convinced of the depth of mathematics and clarity of his exposition. If you do not have a copy of one of Serre’s books on your shelves right now (although I suspect many of you do), one good place to start would be the four volumes of his collected papers entitled Oeuvres and published by Springer Verlag.

These four volumes collect 173 papers starting with 1949’s “Extensions de corps ordonnés” and ending with “La vie et l’oeuvre d’ André Weil” published in 1999. As those titles suggest, most of the papers included are in French, although a good number are in English as well. (It is worth pointing out, however, that Serre’s French is clear and easy to read. In fact, this reviewer actually convinced himself he could read French while in graduate school, when it turns out he could just read Serre’s French) The papers are collected in their original versions (so one can watch the development of mathematical typesetting just by flipping through the pages of Oeuvres, but many papers contain corrections and updated thoughts from Serre himself as new endnotes.

With four volumes and nearly 2500 pages collected in Ouevres, there are clearly many highlights. I am sure that most readers would come up with a different list, but some of mine include:

• A paper published in the Annals in 1953 entitled “Groupes d’homotopie et classes de groupes abéliens”, in which Serre shows (among many other things) that for all even $n$ there is a map from the homotopy group $\pi_i(S^n)$ to the direct sum $\pi_{i-1}(S^{n-1}) \oplus \pi_{i-1}(S^{2n-1})$ whose kernel and cokernel are both finite groups with order a power of two. He also proves that for all $n>2$ we have that $\pi_{n+1}(S^n)$ and $\pi_{n+2}(S^n)$ are cyclic of order two.
• “Géométrie algébrique et géométrie analytique”, the 1956 paper which introduced Serre’s GAGA philosophy connecting analytic geometry to algebraic geometry.
• The 1968 paper with John Tate entitled “Good Reduction of Abelian Varieties” which proves (among other things) that having good reduction is a property of the isogeny class of an abelian variety and there are explicit criteria that one can formulate.
• From 1983, the paper “Sur le nombre des points rationnels d’une courbe algébrique sur un corps fini” gives explicit bounds for the number of points that a curve of a given genus can have that are defined over a fixed finite field.
• “Sur les représentations modulaires de degré 2 de $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$” originally appeared in the Duke Mathematical Journal in 1987, and is the paper in which Serre formulates what became known as “Serre’s Modularity Conjecture” saying that all representations of the absolute Galois group of $\mathbf{Q}$ satisfying certain conditions come about from modular forms. This conjecture is intimately intertwined with Wiles’ proof of Fermat’s Last Theorem.
• Serre’s contributions to the classic book Algebraic Number Theory, edited by Cassels and Frohlich, which essentially introduced Local Class Field Theory to mathematics. This book was long out of print (although not any more), and copies were handed down from graduate student to graduate student, largely due to Serre’s chapters.

I could keep summarizing Serre’s results all day, but you would be far better off if you go track down the original works. And the collected papers in Oeuvres gives an excellent way of doing so. If your tastes in mathematics overlap with Serre’s — and it would be hard not to given his breadth — then I think you will greatly enjoy tracking down a copy of these volumes and looking through them.

Darren Glass is an Associate Professor of Mathematics at Gettysburg College who can be reached at [email protected]. His first exposure to Serre came at the end of his first year of graduate school, when one of his professors at the University of Pennsylvania appeared in his office doorway and said “Darren, this summer you and I are reading Corps Locaux together. You get to decide if we will do it in French or English.” The combination of Serre’s writing and the subsequent discussions with Professor Shatz both broke him out of his first-year-of-grad-school-malaise and convinced him to become an arithmetic geometer.