Correspondence Serre – Tate. Volume I.
(Correspondance Serre – Tate. Volume I.)

*(French, English)*Zbl 1326.11001
Documents Mathématiques 13. Paris: Société Mathématique de France (SMF) (ISBN 978-2-85629-802-2/hbk). xviii, 448 p. (2015).

This first volume of the correspondence between two of the most influential number theorists of the 20th century, Jean-Pierre Serre and John Tate, both recipients of the Abel prize, contains about 200 letters written between 1956 and 1973. Serre met Tate in 1952 in the famous Artin-Tate seminar on class field theory in Princeton and received the Fields Medal in 1954; Tate was a student of Emil Artin and has his name attached to a host of objects that have turned out to be of central importance for the development of number theory in the 20th and 21st centuries (Tate-Shafarevich group, Tate module, Tate cohomology groups, Lubin-Tate theory and many more).

The letters in this volume are written in English and French, and they are filled with mathematics, connected to Serre’s and Tate’s own research as well as to that by their colleagues, such as Grothendieck, Serge Lang, and other mathematicians attached in some way to Bourbaki. It is of course impossible to do justice to the content of these letters, but at least let me mention a few keywords: Galois cohomology, formal groups, Lie groups, profinite groups, representation theory, zeta functions, complex multiplication, elliptic curves, modular forms, and Ramanujan’s \(\tau\) function.

Among the many non-mathematical topics covered in these letters let me mention the recurring theme of Tate being slow in writing down his results, a chess game played between Serre and Tate in 1967 (they followed the world championships, in particular the famous match between Fischer and Spasski in 1972), or Tate’s discovery of the Beatles in 1967: in his letter to Serre he mentions that he has not “tuned in, turned on, and dropped out [\(\ldots\)] to live in the beautiful, colorful psychedelic world of love and dreams”, but that he has “been learning to appreciate the Beatles – they are really great – Serge was right”.

The letters in this volume are written in English and French, and they are filled with mathematics, connected to Serre’s and Tate’s own research as well as to that by their colleagues, such as Grothendieck, Serge Lang, and other mathematicians attached in some way to Bourbaki. It is of course impossible to do justice to the content of these letters, but at least let me mention a few keywords: Galois cohomology, formal groups, Lie groups, profinite groups, representation theory, zeta functions, complex multiplication, elliptic curves, modular forms, and Ramanujan’s \(\tau\) function.

Among the many non-mathematical topics covered in these letters let me mention the recurring theme of Tate being slow in writing down his results, a chess game played between Serre and Tate in 1967 (they followed the world championships, in particular the famous match between Fischer and Spasski in 1972), or Tate’s discovery of the Beatles in 1967: in his letter to Serre he mentions that he has not “tuned in, turned on, and dropped out [\(\ldots\)] to live in the beautiful, colorful psychedelic world of love and dreams”, but that he has “been learning to appreciate the Beatles – they are really great – Serge was right”.

Reviewer: Franz Lemmermeyer (Jagstzell)

##### MSC:

11-03 | History of number theory |

01A60 | History of mathematics in the 20th century |

01A70 | Biographies, obituaries, personalia, bibliographies |

01A72 | Schools of mathematics |

14-03 | History of algebraic geometry |