8 ALESSANDRO FIGA-TALAMANCA

The modular function

laiF

attains the values

qk

with k

E

Z.

In other words

laiF

=

qk

if

a

E

p-ko

and

a

tJ_

p-k+

1

o.

The closed balls of radius

qk

are exactly the translates

p-ko

+

a

of the ball

p-ko.

Two such balls

p-kO+a

and

p-kO+b

are disjoint if

la-biF qk

and they

coincide if

Ia-

biF

~

qk.

Thus the partition ofF into closed balls of radius

qk

is

the same as the partition of F associated to the equivalence relation

We can now proceed to the definition of the tree associated to F.

We let X, the set of vertices, to be the set of closed balls ofF, that is the set

X=

{pkO

+a : k

E

Z, a

E

F}.

We say that two vertices

pkO +a

and

phO

+

b

are adjacent if

lh- kl

=

1 and

one of the two balls is contained in the other. In other words if either k

=

h

+

1

and

a-bE phO

or else h

=

k

+

1 and

a-bE pkO.

It is not difficult to see that with this definition of edges X becomes a tree.

But we can observe that every ball of radius

qk

is contained in exactly one ball of

radius

qk+

1

and contains exactly

q

balls of radius

qk-l.

This means that the tree

is homogeneous of degree q

+

1.

Fig.1 represents now the tree of a local field in the case q

=

2. The leftmost

balls are

pkO

as indicated in the figure. The balls of a given radius are arranged in

rows. We shall return later on on the geometric significance of these rows.

We shall presently identify the boundary of X. First of all let us consider the

infinite chain

p-ko,

k

=

0, 1, 2, .... This chain identifies a point of the boundary

which we shall denote by oo. Every other chain starting at the vertex

0

will

eventually consist of decreasing balls (Fig.1).

In other words the boundary points

may be identified as the limits

00 00

limsUpXk

=

n u

Xk,

n=l k=n

where x

0

,

x

1

, ...

is a chain of balls starting at x

0

=

0.

The above limits are either

the field itself F (when

Xk

is an increasing sequence of balls) or else the singleton

{a}

with

a

E

F, when the chain

x

0

, Xk, ...

is eventually decreasing. In the first case

the boundary point identified by the chain

Xk

is oo, in all other cases is an element

a of F. Accordingly we may identify the boundary of the tree with F

U {

oo }, the

boundary point oo being associated to the chain (p-kO)k'=o·

Notice that balls of equal radius have been arranged in rows in Fig.l. This

arrangement may be given an interpretation in terms of the geometry of the tree,

independently of its graphic representation.

Let's go back for a moment to the case of a general homogeneous tree

X,

with

boundary

n.

Fix a point w E

n.

If

X

and y are two vertices consider the infinite

chains

[x,w)

and

[y,w).

They intersect in a geodesic

[z,w)

=

[x,w)

n

[y,w).

That

is

z

is the first vertex of

[x,

w)

and

[y,

w)

which belongs to both chains. Define now

the relation

xRy

if

d(x, z)

=

d(y, z).

The modular function

laiF

attains the values

qk

with k

E

Z.

In other words

laiF

=

qk

if

a

E

p-ko

and

a

tJ_

p-k+

1

o.

The closed balls of radius

qk

are exactly the translates

p-ko

+

a

of the ball

p-ko.

Two such balls

p-kO+a

and

p-kO+b

are disjoint if

la-biF qk

and they

coincide if

Ia-

biF

~

qk.

Thus the partition ofF into closed balls of radius

qk

is

the same as the partition of F associated to the equivalence relation

We can now proceed to the definition of the tree associated to F.

We let X, the set of vertices, to be the set of closed balls ofF, that is the set

X=

{pkO

+a : k

E

Z, a

E

F}.

We say that two vertices

pkO +a

and

phO

+

b

are adjacent if

lh- kl

=

1 and

one of the two balls is contained in the other. In other words if either k

=

h

+

1

and

a-bE phO

or else h

=

k

+

1 and

a-bE pkO.

It is not difficult to see that with this definition of edges X becomes a tree.

But we can observe that every ball of radius

qk

is contained in exactly one ball of

radius

qk+

1

and contains exactly

q

balls of radius

qk-l.

This means that the tree

is homogeneous of degree q

+

1.

Fig.1 represents now the tree of a local field in the case q

=

2. The leftmost

balls are

pkO

as indicated in the figure. The balls of a given radius are arranged in

rows. We shall return later on on the geometric significance of these rows.

We shall presently identify the boundary of X. First of all let us consider the

infinite chain

p-ko,

k

=

0, 1, 2, .... This chain identifies a point of the boundary

which we shall denote by oo. Every other chain starting at the vertex

0

will

eventually consist of decreasing balls (Fig.1).

In other words the boundary points

may be identified as the limits

00 00

limsUpXk

=

n u

Xk,

n=l k=n

where x

0

,

x

1

, ...

is a chain of balls starting at x

0

=

0.

The above limits are either

the field itself F (when

Xk

is an increasing sequence of balls) or else the singleton

{a}

with

a

E

F, when the chain

x

0

, Xk, ...

is eventually decreasing. In the first case

the boundary point identified by the chain

Xk

is oo, in all other cases is an element

a of F. Accordingly we may identify the boundary of the tree with F

U {

oo }, the

boundary point oo being associated to the chain (p-kO)k'=o·

Notice that balls of equal radius have been arranged in rows in Fig.l. This

arrangement may be given an interpretation in terms of the geometry of the tree,

independently of its graphic representation.

Let's go back for a moment to the case of a general homogeneous tree

X,

with

boundary

n.

Fix a point w E

n.

If

X

and y are two vertices consider the infinite

chains

[x,w)

and

[y,w).

They intersect in a geodesic

[z,w)

=

[x,w)

n

[y,w).

That

is

z

is the first vertex of

[x,

w)

and

[y,

w)

which belongs to both chains. Define now

the relation

xRy

if

d(x, z)

=

d(y, z).