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).
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