Figure 1
spaces was available since it was translated and published in the Soviet Union (even
before it was published in the West). It was Tate’s theory of affinoid algebras and
affinoid domains that compensated for the lack of the usual complex analytic tools
in the non-archimedean world. I studied Tate’s work intensively and adjusted to
the new framework, introducing a category of analytic spaces which eventually
coincided with that given above. In the present framework, it is precisely the full
subcategory of the category of analytic spaces consisting of the spaces without
boundary. They are glued from the interiors of affinoid spaces (the interior of
M(A) consists of the points x for which the corresponding character A H(x)
is a completely continuous operator), and include the analytifications of algebraic
varieties. The new analytic spaces were applied to define the common spectrum
of a finite family of commuting bounded linear operators, to develop holomorphic
functional calculus, and to prove the Shilov idempotent theorem. The latter states
that for any open and closed subset of the spectrum M(A) of a commutative Banach
algebra A there exists a unique idempotent e A which is equal to 1 precisely at
that subset.
At that time I found that the process of writing down of what I was starting
to understand was very enjoyable and extremely helpful for better understanding.
The need to express an idea forced me to concentrate on each small object or
detail of reasoning. This concentration helped me see hidden and refined nuances
which could change the whole picture, or to discover again and again a deep-rooted
prejudice or wrong vision or simple stupidity.
The first typewritten text was finished in April of 1986, and I succeeded in
passing it to Professor Barry Mazur, who knew me from my previous work. Later
on, I was surprised to learn that my text had been accepted by the American
Mathematical Society for publication as a book. But I was actually lucky that
everybody at AMS immediately forgot about me finishing that book, and so I
could continue to rewrite the text infinitely many times, gradually extending the
framework of the new analytic spaces and investigating their amazing properties.
Tate’s paper was still the only source of my knowledge in rigid analytic geom-
etry, when I considered the following situation. Assume that the ground field k is
algebraically closed and the characteristic of its residue field is not 2, and let E be
an elliptic curve over k defined by the affine equation y2 = x(x 1)(x λ) with
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