Algebraicity theorems in Diophantine geometry following J.-B. Bost, Y. André, D. & G. Chudnovsky. (Théorèmes d’algébricité en géométrie diophantienne d’après J.-B. Bost, Y. André, D. & G. Chudnovsky.) (French) Zbl 1044.11055

Chambert-Loir, Antoine

Bourbaki seminar. Volume 2000/2001. Exposés 880-893. Paris: Société Mathématique de France (ISBN 2-85629-130-9/pbk). Astérisque 282, 175-209, Exp. No. 886 (2002).

This Bourbaki seminar primarily describes the work of J.-B. Bost [Publ. Math., Inst. Hautes Étud. Sci. 93, 161–221 (2001; Zbl 1034.14010)], who showed the following. Let \(K\) be a number field embedded in \(\mathbb C\), let \(X\) be a smooth algebraic variety over \(K\) (i.e., an integral separated scheme of finite type over \(K\)), and let \(F\) be an algebraic subbundle of the tangent bundle \(T_X\). We assume that \(F\) is involutive; i.e., closed under the Lie bracket. Then \(F\) defines a holomorphic foliation of the complex manifold \(X(\mathbb C)\). Bost showed that the leaf \(\mathcal F\) through a rational point \(P\in X(K)\) is algebraic if the following local conditions are satisfied: (i) for almost all prime ideals \(\mathfrak p\) of the ring of integers \(\mathcal O_K\) of \(K\), the \(p\)-curvature of the reduction modulo \(\mathfrak p\) of the subbundle \(F\subseteq T_X\) vanishes at \(P\) (here \(p\) is the prime of \(\mathbb Z\) lying below \(\mathfrak p\)); and (ii) the manifold \(\mathcal F\) satisfies the Liouville property: every plurisubharmonic function on \(\mathcal F\) bounded from above is constant. For example, \(\mathcal F\) satisfies the Liouville property if it is a holomorphic image of a complex algebraic variety minus a closed analytic subset.
The article also shows how Bost’s theorem is related to a conjecture of Grothendieck, predicting when a linear system \((d/dz)Y=A(z)Y\), \(A(z)\in M_d(\mathbb Q(z))\), has a basis of algebraic solutions. In addition, the article shows how Bost’s theorem implies a theorem of Y. André giving a local criterion for when a differential form on a smooth variety is exact; a local criterion for two elliptic curves to be isogenous (special case of a theorem of Faltings); and even a theorem of Kronecker stating that if \(\alpha\) is an element of a number field \(K\) such that \(\alpha\) is congruent to an element of \(\mathbb F_p\) for all but finitely many primes \(\mathfrak p\) of \(K\), then \(\alpha\in\mathbb Q\).
For the entire collection see [Zbl 1007.00024].

Reviewer: Paul Vojta (Berkeley)

MSC:

11G35	Varieties over global fields
11J81	Transcendence (general theory)
14G40	Arithmetic varieties and schemes; Arakelov theory; heights

Keywords:

algebraicity; foliation; Hermitian vector bundle; \(p\)-curvature; slope; Grothendieck conjecture; Arakelov geometry; Liouville property

Citations:

Zbl 1034.14010