NON-ARCHIMEDEAN ANALYTIC GEOMETRY: FIRST STEPS 5

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