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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