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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
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].

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