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Next: References Up: Diophantine Equations and Diophantine Previous: The Subspace Theorem

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]).

next up previous
Next: References Up: Diophantine Equations and Diophantine Previous: The Subspace Theorem