Grothendieck's “Long March through Galois theory”

Summary

Note. This short text was originally written as a contribution to the “Grothendieck day” which took place in Utrecht on April 12, 1996. It is brief and informal, and was intended to give the audience some very partial idea of what is contained in Grothendieck's long manuscript “La Longue Marche à travers la Théorie de Galois”. The close connections between the ideas expressed there and those in parts of the Esquisse and the letter to Faltings in this volume make it relevant to publish it here.

Alexander Grothendieck wrote the Long March between January and June 1981. It consists of about 1600 manuscript pages, and nearly as much again in various addenda and developments. About the first 600 pages, consisting of §§1-37, have been read and edited; the main body of the Long March consists of §§1-53. From the table of contents of §§38-53, it seems that they are mainly devoted to the close study of the ideas expressed in the first part, in the cases where (g, n) = (0,3) and (1,1). In 1984, Grothendieck wrote and distributed the Esquisse d'un Programme, a 57-page text part of which summarizes, sometimes in a more advanced form, the main themes and problems considered in the Long March. Both texts are devoted to raising deep questions and examining various approaches to them, and contain few explicit theorems.