So far, we have seen how -dimensional infrastructures can be defined. In the case of one-dimensional infrastructures, we saw that there is a (more or less) obvious way how to define a reduction map, which does not extend to the
-dimensional case. We next want to motivate how a reduction map can be defined given
and
, using additional information which might be easier to obtain.
First, introduce on a lexicographic order as follows: for
and
, define
Note that this choice is rather random and can easily be replaced by other choices.
Assume that is a lattice,
a finite set and
injective. Consider the projection
,
and
. Defining a function
is the same as defining a function
which is invariant under
, i.e. satisfies
for all
; in that case, we can set
. Note that the condition
translates to
.
Hence, we have a discrete set which is invariant under translation by
, and we want to define a function
satisfying
.
Both of the two sections which follow describe one way to obtain such and
. The way describes in the second section fits perfectly for all totally real number fields
: think of
as the image of the ring of integers
under all embeddings
, i.e.
The first section resembles more the general global field situation. The set will consist of a finite set of ideals with bounded norms. The degree map will be the logarithm of the norm, and the
's correspond to the degrees of the infinite places.
Constructing a Reduction Map.
In this section, we describe a way to construct a reduction map , given
.
The main idea in the following is that if we want to define for
, to consider the area
and look at all elements . By adding additional (numeric) information to every of these elements, one obtains an order (by comparing the additional information) which hopefully has a largest element, or a finite set of largest elements. From these largest elements, one chooses the largest one with respect to the lexicographic order
as
.
To make this “additional information” more precise, we consider special functions which should behave in a good way:
A function is said to be reduction-inducing if
-
there exist real numbers
such that, for
and
, we have
and
-
for every
, we have
Note that by this definition, there exist such that
for all . Moreover, note that these functions with
fixed correspond to functions
by
for .
Let be a reduction-inducing function. For
and
, consider
Note that since for
, and
, we see that
is finite for every choice of
. If
, we have
, and as
we get
for
. Hence,
exists. Then, define
.
Let be a constant such that for all
, we have
Note that since is reduction-inducing, a maximal such
exists.
For , we have
. For any
, if
, we have
and
. In fact,
.
For the first statement, it suffices to show . But note that if
, we would have
and hence
, a contradiction.
For the second statement, note that . Moreover,
for
, whence we get
. This shows the inequality on
. Now clearly
, whence
. Now
, whence
. As
.
Using Minima of Lattices.
In this section, we describe how to obtain and
from an
-dimensional lattice
. We require that for every
, we either have
for
for all
. More precisely, consider the map
,
. We assume that there exists a constant
with
for all
.
In fact, one can replace by any discrete subset with some additional properties which give similar results as Minkowski's Lattice Point Theorem.
A minimum of is an element
such that for all
with
for all
, we either have
or
for all
. Denote the set of all minima by
.
First, we will show that such minima exist:
Let . Then there exists a minimum
of
with
for all
.
This follows from the fact that is discrete. For
, define
As is discrete,
is always finite.
In particular, is finite. Assume that
is not a minimum (in which case we could choose
). Then there exists some
with
for some
. In that case,
. Now either
is a minimum, in which case we choose
, or it is not. In that case, we can repeat the procedure with
instead of
. As the size of these sets decreases every step and the sets are finite but non-empty, we eventually must find some
which is a minimum.
Define on
by
and consider the map
First, if, and only if,
. Let
Let . Then, there exists some
with
and
. In particular,
.
Here, is the determinant of the lattice
, i.e. the volume of one fundamental parallelepiped of
.
For , consider the set
By Minkowski's Lattice Point Theorem, we have for
i.e.
Since is closed and
discrete, a limit argument shows that this also holds for
. By the previous lemma, there exists a minimum
of
which lies in
; let
. Now
for
as
, whence
.
Now , whence
. Now
, whence
. Therefore, we get
i.e. .
Define ; this is a discrete subgroup of
. We assume that
is a lattice in
, i.e. contains a basis of
. We can define
and
such that
, if
,
is the projection. To get an
-dimensional infrastructure, we are left to define
.
For that, we proceed as in the proof of the second lemma in this section. For , consider
Let be minimal with
, and let
. Then
satisfies the properties in the statement of the lemma, i.e. lies near to
itself. Moreover, one quickly checks that
for all
.
Hence, we obtain an -dimensional infrastructure.
Comments.