NON-ARCHIMEDEAN ANALYTIC GEOMETRY: FIRST STEPS 3

to view the space

A1

as the aﬃne line was the fact that it provided an answer to

the question on the spectrum of a bounded linear operator I posed to myself at the

very beginning. Namely, the spectrum of such an operator is a nonempty compact

subset of A1, and it coincides with the complement of the analyticity set of its

resolvent. The field k is naturally embedded in the aﬃne line as a dense subset,

and the operators studied by Misha Vishik were precisely those with spectrum

contained in k.

It was a pleasant exercise to extend Vishik’s results to arbitrary operators, and

it helped me to understand better the topological tree-like structure of the non-

archimedean aﬃne and projective lines, to get used to them, and to accept them

as reality. During this work I met with Misha several times to tell him about the

progress. At that time, he was the only person (besides my wife) who shared my

excitement about all this. The usual reaction of my colleagues was simple indiffer-

ence at best, and the quite understandable reason for that was nicely expressed by

Professor Manin. When I told him about what I was doing, he observed that it is

worthwhile to develop a general theory only having in mind a concrete problem. Of

course, understanding what the spectrum of a non-archimedean operator should be

was not a concrete problem. Had I followed this wise advice, I would have turned

back to concrete problems I had in abundance in the area of computers, and would

probably have become rich during the present age of the high tech boom since I

was a really good programmer. Fortunately, I was already stupid enough to miss

such an attractive opportunity, and I continued my exploration of the unknown

new world revealed to me by a fluke.

My job occupied me five days per week from 8am till 5pm. It took me several

years to learn to devote an hour or two to mathematics during working hours.

Time free from my job belonged to my family, and when I was completely hooked

on mathematics and an hour or two per day was not enough for it, I discovered an

additional source of time. I learned to get up every day very early (often as early

as at 2am), and thus extended the time for doing mathematics. At this time of the

day, the world around me was quiet and fresh, nobody and nothing disturbed me,

my head was clear, and I could plunge into another world to explore and describe

it.

When I had finished writing everything I had in mind, I could look at it quietly

and listen to an inner feeling that something was not satisfactory. I thought about

this from time to time more and more often, but could not even express what

tormented me. One day at the very end of 1985 all this obsessed me. I could not

stop thinking about it at my job, and later at home. I did not go to bed early

as usual. The right question and an immediate answer to it came early the next

morning.

As I mentioned above, the aﬃne line

A1

is provided with a sheaf of rings OA1 .

Its stalk OA1,x at a point x is a local ring with residue field κ(x) provided with a

valuation that extends that on k. If x is of type (1) (i.e., corresponds to an element

of k), then OA1,x is the algebra of convergent power series at that point, and so

κ(x) = k. Otherwise, it is a field of infinite degree over k, and so it coincides with

the non-complete field κ(x). If H(x) denotes the completion of κ(x), one gets a

character k[T ] → H(x) over k. The question that came to me on that early morning

was the following. What are all possible characters k[T ] → K to non-archimedean

fields K over k?