We mention some recent developments in Diophantine geometry which are related to the results from the second section. This section is more specialized.

We write
for the multiplicative group
with
coordinatewise
multiplication
.
The group
is the group of complex points of a
group variety
, called the -dimensional linear torus.
Lang ([14], p. 220)
proposed the following conjecture:
*Let be either
or an abelian variety defined over
.
Let be a subgroup of
of finite rank (i.e., has a
finitely
generated subgroup such that
is a torsion group).
Further, let be an algebraic subvariety of defined over
and let
denote the exceptional set of , that is the union of all
translates
of positive dimensional algebraic subgroups of which are contained
in .
Then the intersection
is finite.*

For instance, if
and is a hyperplane given by
then
is the set of solutions of
in
, that is, we
have an equation
of type (2). The non-degenerate solutions of this equation
(i.e., with non-vanishing subsums) are precisely the points in
. So Lang's conjecture implies that
(2)
has only finitely many non-degenerate solutions.

Let be a projective curve of genus
defined over an algebraic number field , let be the Jacobian of
,
and let
. We assume that
. We know that
and that is finitely generated (the Mordell-Weil Theorem).
Thus Lang's conjecture implies Mordell's conjecture that is
finite.

In the 1980's, Laurent [15] proved Lang's conjecture in the case
that
. Laurent's proof was based on the p-adic Subspace Theorem.
In 1983, Faltings [9] proved Mordell's conjecture. Unlike
Laurent,
Faltings did not use Diophantine approximation. In 1991,
Vojta [29] gave a totally different proof of Mordell's
conjecture
based on Diophantine approximation.
Then by extending Vojta's ideas to
higher dimensions,
Faltings [10],[11]
achieved the following breakthrough, which almost settled Lang's
conjecture for
abelian varieties:
*Let be an abelian variety, and let be a projective
subvariety of , both defined over an algebraic number field .
Then
is finite.*

Subsequently, the proof of Lang's conjecture was completed by McQuillan
[18]. We refer to the books [12], [2] for an
introduction.

Very recently, Rémond proved the following remarkable quantitative
version
of Lang's conjecture.
Rémond used Faltings' arguments,
but he managed to simplify them considerably.

If
we assume that
by identifying
with the point
.
if is an abelian variety we assume that
is contained in some projective
space
and that the line sheaf is symmetric.
Further we assume that is defined over the field of algebraic
numbers.
In both cases, has dimension , is an algebraic
subvariety of of dimension and degree (with respect to the
embeddings
chosen above) defined over the algebraic numbers,
and is a subgroup of
of finite rank .
**Theorem (Rémond).**
*(i) Let
. Then
has cardinality at
most
([21]).
(ii) Let be an abelian variety. Then
has cardinality at most
,
where is an effectively computable constant depending on
([19],[20]).
*