Diophantine geometry

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 ${\mathbb{G}}_m^n({\mathbb{C}})$ for the multiplicative group $({\mathbb{C}}^*)^n$ with coordinatewise multiplication $(x_1,\ldots,x_n)(y_1,\ldots,y_n)=(x_1y_1,\ldots,x_ny_n)$. The group ${\mathbb{G}}_m^n({\mathbb{C}})$ is the group of complex points of a group variety ${\mathbb{G}}_m^n$, called the $n$-dimensional linear torus. Lang ([14], p. 220) proposed the following conjecture:
Let $A$ be either ${\mathbb{G}}_m^n$ or an abelian variety defined over ${\mathbb{C}}$. Let $\Gamma$ be a subgroup of $A({\mathbb{C}})$ of finite rank (i.e., $\Gamma$ has a finitely generated subgroup $\Gamma _0$ such that $\Gamma /\Gamma _0$ is a torsion group). Further, let $X$ be an algebraic subvariety of $A$ defined over ${\mathbb{C}}$ and let $Z(X)$ denote the exceptional set of $X$, that is the union of all translates of positive dimensional algebraic subgroups of $A$ which are contained in $X$. Then the intersection $(X\backslash Z(X))\cap\Gamma$ is finite.
For instance, if $A={\mathbb{G}}_m^n$ and $X$ is a hyperplane given by $a_1x_1+\cdots+a_nx_n=1$ then $X({\mathbb{C}})\cap\Gamma$ is the set of solutions of $a_1x_1+\cdots+a_nx_n=1$ in $(x_1,\ldots,x_n)\in\Gamma$, 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 $(X\backslash Z(X))\cap\Gamma$. So Lang's conjecture implies that (2) has only finitely many non-degenerate solutions.
Let $X$ be a projective curve of genus $\geqslant 2$ defined over an algebraic number field $K$, let $A$ be the Jacobian of $X$, and let $\Gamma =A(K)$. We assume that $X\subset A$. We know that $Z(X)=\emptyset$ and that $A(K)$ is finitely generated (the Mordell-Weil Theorem). Thus Lang's conjecture implies Mordell's conjecture that $X(K)$ is finite.

In the 1980's, Laurent [15] proved Lang's conjecture in the case that $A={\mathbb{G}}_m^n$. 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 $A$ be an abelian variety, and let $X$ be a projective subvariety of $A$, both defined over an algebraic number field $K$. Then $(X\backslash Z(X))(K)$ 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 $A={\mathbb{G}}_m^n$ we assume that $A\subset{\mathbb{P}}^n$ by identifying $(x_1,\ldots,x_n)\in{\mathbb{G}}_m^n$ with the point $(1,x_1,\ldots,x_n)\in{\mathbb{P}}^n$. if $A$ is an abelian variety we assume that $A$ is contained in some projective space ${\mathbb{P}}^N$ and that the line sheaf $O(1)$ is symmetric. Further we assume that $A$ is defined over the field of algebraic numbers. In both cases, $A$ has dimension $n$, $X$ is an algebraic subvariety of $A$ of dimension $m$ and degree $d$ (with respect to the embeddings chosen above) defined over the algebraic numbers, and $\Gamma$ is a subgroup of $A(\overline{{\mathbb{Q}}})$ of finite rank $r$.
Theorem (Rémond). (i) Let $A={\mathbb{G}}_m^n$. Then $(X\backslash Z(X))\cap\Gamma$ has cardinality at most $(2d)^{n^2(m+1)^{4m^2}(r+1)}$ ([21]).
(ii) Let $A$ be an abelian variety. Then $(X\backslash Z(X))\cap\Gamma$ has cardinality at most
$\big( c_A\cdot d)^{n^{5(m+1)^2}(r+1)}$, where $c_A$ is an effectively computable constant depending on $A$ ([19],[20]).