Book review: The Serre–Tate correspondence

For the past month I’ve been punctuating my life by reading the correspondence between Jean-Pierre Serre and John Tate, recently published in two volumes. Anyone interested in the development of number theory and algebraic geometry will find something to enjoy here.

The book was presumably suggested by the success of the Grothendieck–Serre correspondence, published by the Société Mathématique de France in 2001 and in English translation by the American Mathematical Society in 2003. The Grothendieck–Serre correspondence, beyond its outstanding mathematical interest, has the additional personal fascination of Grothendieck’s story. At first a complete outsider to algebraic geometry, he becomes the master builder of the subject in the 1960s, before rejecting mathematics and, by the end, the rest of humanity.

By comparison, Serre and Tate are reasonable men. The attraction of their correspondence lies in the mathematical ideas that they gradually develop, over the years from 1956 to 2009. Some of the key topics are Galois cohomology (essentially created by Serre and Tate), Tate’s notion of rigid analytic spaces, the Tate conjecture on algebraic cycles, Tate’s invention of p-adic Hodge theory, and Serre’s work on the image of Galois representations, for example for elliptic curves.

Serre usually writes in French, and Tate in English; but both writers make occasional use of the other language for the fun of it.

One running theme is Tate’s reluctance to write up or publish some of his best work. Serre encourages Tate and edits Tate’s papers, but sometimes has to concede defeat. Mazur and Serre started to prepare the publication of Tate’s Collected Papers in about 1990, which would include letters and unpublished work; sadly, nothing has appeared. Serre reports that the AMS has revived the project, and concludes: “I cross my fingers.”

A major topic of the correspondence starting in the 1970s is the relation between modular forms and Galois representations. Deligne and Serre showed in 1974 that a modular form of weight 1 determines a Galois representation with image a finite subgroup of PGL(2,C). At that time, however, it was a serious computational problem to give any example at all of a modular form of weight 1 for which the image is an “interesting” subgroup (that is, A4, S4, or A5, not a cyclic or dihedral group). Tate and a group of students found the first example on June 21, 1974. Soon Tate becomes fascinated with the HP25 programmable calculator as a way to experiment in number theory.

Both Serre and Tate are strongly averse to abstract theories unmoored to explicit examples, especially in number theory. This is a very attractive attitude, but it had one unfortunate effect. One of Serre’s best conjectures, saying that odd Galois representations into GL(2) of a finite field come from modular forms, was formulated in letters to Tate in 1973. But for lack of numerical evidence, Serre ended up delaying publication until 1987. The conjecture played a significant role in the lines of ideas leading to Wiles’s proof of Fermat’s last theorem. Serre’s Conjecture was finally proved by Khare and Wintenberger.

Finally, the correspondence has its share of mathematical gossip. One memorable incident is the Fields Medals of 1974. Tate is on the Fields Medal committee, and Serre suggests “Manin-Mumford-Arnold” as not a bad list, with Arnold as the strongest candidate outside number theory and algebraic geometry. In the event, the award went only to two people, Bombieri and Mumford. At least in the case of Arnold, it seems clear (compare this MathOverflow question) that this was a disastrous result of official anti-Semitism in the USSR, with the Soviet representative to the International Mathematical Union, Pontryagin, refusing to allow the medal to go to Arnold.

I hope that some mathematical readers will go on from the Serre–Tate correspondence to Serre’s Collected Papers. Serre took the idea of cohomology from topology into algebraic geometry and then into number theory. He is one of the finest writers of mathematics. I recommend his papers without reservation.

Correspondance Serre–Tate, 2 volumes. Editée par Pierre Colmez et Jean-Pierre Serre. Société Mathématique de France (2015).

Photo was from the Cambridge branch of Cats Protection, but a different cat is now featured.