6 NON-ARCHIMEDEAN ANALYTIC GEOMETRY: FIRST STEPS

Figure 2

0 |λ| ≤ 1 and |λ − 1| = 1. (The latter can be always achieved after a change of

coordinates.) Recall that the projective line

P1

has the property that any two differ-

ent points can be connected by a unique path. Let us connect the points 0, 1, λ and

∞ in

P1.

We get one of the two graphs Γλ presented in Figure 1, which correspond

to the cases |λ| = 1 and |λ| 1. (For brevity the point p(E(0; r)) is denoted by pr.)

The complement of Γλ in

P1

is a disjoint union of open discs of the form D(a; ra)

with a ∈ k\{0, 1, λ} and ra = min{|a|,|a − 1|,|a − λ|}. Every such disc is glued to

its boundary which is a point of Γλ, and Γλ is a strong deformation retraction of

the whole projective line

P1.

Consider now the x-projection π :

Ean

→

P1

from

the analytification

Ean

of E. Since the characteristic of the residue field of k is not

two, the square root of each of the linear factors of x(x −1)(x− λ) can be extracted

at D(a; ra) and, therefore, the preimage of

π−1(D(a;

ra)) is a disjoint union of two

open discs which are glued to their boundaries at the preimage π−1(Γλ). Thus, the

latter is a strong deformation retraction of Ean. If 0 r |λ|, then the square roots

of x − λ and x − 1 are extracted at the open annulus D(0; r + ε)\E(0; r − ε) with

0 r − ε r + ε |λ|, but the square root of x is not. This means that each point

of the interval that connects 0 with p|λ| has a unique preimage in Ean. Similarly,

each point from the intervals that connect 1 with p1, λ with p|λ|, and ∞ with p1,

has a unique preimage in

Ean.

In particular, if |λ| = 1, then

π−1(Γλ)

∼

→ Γλ. If now

|λ| 1, then the square roots of x − 1 and of the product x(x − λ) are extracted

at the open annulus D(0; r + ε)\E(0; r − ε) with |λ| r − ε r + ε 1. This

means that each point pr with |λ| r 1 has two preimages in

Ean,

and the graph

π−1(Γλ)

has the form presented in Figure 2.

Thus, the analytic curve

Ean

is contractible if |λ| = 1, and homotopy equivalent

to a circle if |λ| 1. It is well known that these two cases correspond to those

when the modular invariant j(E) is integral or not. But the latter case |j(E)| 1

is precisely that of a Tate elliptic curve. Wow, such a curve is homotopy equivalent

to a circle! I was always fascinated by Tate elliptic curves, but never understood

the reason for which they admit uniformization. And here I had a very elementary

explanation of this astonishing phenomenon discovered by Tate; it reminded me of

the classical construction of the Riemann surface of an algebraic function.

Of course, all this strongly lifted up my spirit and eagerness in exploration of

the new spaces. This was very timely since it distracted me from a serious health

problem I had at that time, not to mention my job in a dull institution and the

reality of a country in an advanced stage of decaying.