Non-archimedean analytic geometry: first steps

Vladimir Berkovich

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 non-

archimedean 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

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http://dx.doi.org/10.1090/ulect/045/01