We mention some recent developments in Diophantine geometry which are related to the results from the second section. This section is more specialized.
for the multiplicative group
is the group of complex points of a
, called the -dimensional linear torus.
Lang (, 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  proved Lang's conjecture in the case
. Laurent's proof was based on the p-adic Subspace Theorem.
In 1983, Faltings  proved Mordell's conjecture. Unlike
Faltings did not use Diophantine approximation. In 1991,
Vojta  gave a totally different proof of Mordell's
based on Diophantine approximation.
Then by extending Vojta's ideas to
achieved the following breakthrough, which almost settled Lang's
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 . We refer to the books ,  for an introduction.
Very recently, Rémond proved the following remarkable quantitative
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 ().
(ii) Let be an abelian variety. Then has cardinality at most
, where is an effectively computable constant depending on (,).