Non-archimedean analytic geometry: first steps

Abstract

When Dinesh Thakur asked me to write an introduction to this volume, I carelessly agreed. Later I started thinking that a short description of my journey to non-archimedean analytic geometry and of some of the circumstances accompanying it might be an entertaining complement to the notes of Matthew Baker and Brian Conrad. Since I had no other ideas, I’ve written what is presented below. I start by briefly telling about myself. I was very lucky to be accepted to Moscow State University for undergraduate and, especially, for graduate studies in spite of the well-known Soviet policy of that time towards Jewish citizens. I finished studying in 1976, and got a Ph.D. the next year. (My supervisor was Professor Yuri Manin.) Getting an academic position would be too much luck, and the best thing I could hope for was the job of a computer programmer at a factory of agricultural machines and, later, at the institute of information in agriculture. As a result, I practically stopped doing mathematics, did not produce papers, and was considered by my colleagues as an outsider. It took me several years to become an expert in computers and nearby fields, and to learn to control my time. Gradually I started doing mathematics again, and my love for it blazed up with new force and became independent of surrounding circumstances. By the time my story begins, I was hungry for mathematics as never before. Thus, my story begins one July evening of 1985 in a train in which I was returning to Moscow after having visited my numerous relatives in Gomel, Belarus. Instead of talking to people near me — my usual occupation during long train trips — I opened a book on classical functional analysis by Yuri Lyubich which my eldest brother Yakov had given me a couple of days before. The basic material of the book was familiar to me. Nevertheless, I was thrilled to read about it again and, suddenly, asked myself: what is the analog of all this over a non-archimedean field k? In particular, what is the spectrum of a bounded linear operator acting on a Banach space over k? It did not take much time to find that, if one defines the spectrum in the same way as in the classical situation, it may be empty even if k is algebraically closed. Indeed, if K is a non-archimedean field larger than k, then the multiplication by any element of K which does not lie in k is an operator with empty spectrum. That such a larger field always exists is easily seen: one can take the completion of the field of rational functions in one variable over k with respect to the Gauss norm. I was very intrigued, and decided to understand what all this meant. I knew that my fellow Manin student Misha Vishik had written a paper on nonarchimedean spectral theory. The next day, I found the paper and started reading it. It turned out Vishik was studying bounded linear operators on a Banach space