Algebraic leaves of algebraic foliations over number fields

@article{Bost2001AlgebraicLO,
title={Algebraic leaves of algebraic foliations over number fields},
author={Jean-Beno{\^i}t Bost},
journal={Publications Math{\'e}matiques de l'Institut des Hautes {\'E}tudes Scientifiques},
year={2001},
volume={93},
pages={161-221}
}
• J. Bost
• Published 1 September 2001
• Mathematics
• Publications Mathématiques de l'Institut des Hautes Études Scientifiques
Summary — We prove an algebraicity criterion for leaves of algebraic foliations defined over number fields. Namely, consider a number field K embedded in C, a smooth algebraic variety X over K, equipped with a K-rational point P, and F an algebraic subbundle of the its tangent bundle TX, defined over K. Assume moreover that the vector bundle F is involutive, i.e., closed unter Lie bracket. Then it defines an holomorphic foliation of the analytic mainfold X(C), and one may consider its leaf…
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References

SHOWING 1-10 OF 45 REFERENCES
Cohomological Dimension of Algebraic Varieties
Let X be a scheme of finite type over a field k. The cohomological dimension of X is the smallest integer n > 0 such that H'(X, F) = 0 for all i > n, and for all quasi-coherent sheaves F on X. There
FORMAL GROUPS AND ZETA-FUNCTIONS
Cl be an elliptic curve over the rational number field Q, uniformized by automorphic functions with respect to some congruence modular group T0(N). In the language of formal groups results of Eichler
Formal groups and the isogeny theorem
In this paper, we prove an isogeny criterion for abelian varieties that involves conditions on the formal groups of the varieties (see Theorem 1.1). In the particular case of abelian varieties overQ
Formal subgroups of abelian varieties
Abstract.In this paper, we generalize the result of [12] in the following sense. Let A be an abelian variety over a number field k, let ? be the Néron model of A over the ring of integers Ok of k.
Heights of projective varieties and positive Green forms
• Mathematics
• 1994
Using arithmetic intersection theory, a theory of heights for projective varieties over rings of algebraic integers is developed. These heights are generalizations of those considered by Weil,
Distribution of Values of Holomorphic Mappings
A vast literature has grown up around the value distribution theory of meromorphic functions, synthesized by Rolf Nevanlinna in the 1920s and singled out by Hermann Weyl as one of the greatest
Notions of Convexity
The first two chapters of the book are devoted to convexity in the classical sense, for functions of one and several real variables respectively. This gives a background for the study in the
A conjecture in the arithmetic theory of differential equations
This article discusses a conjectural description of the Lie algebra of the differential Galois group attached to a linear differential equation as being the smallest algebraic Lie algebra whose
G-functions and geometry
This is an introduction to some geometrie aspects of G-function theory, something intermediate between a standard monograph and a research artic1e; it is not a complete survey of the topic.
Arithmetic moduli of elliptic curves
• Mathematics
• 1985
This work is a comprehensive treatment of recent developments in the study of elliptic curves and their moduli spaces. The arithmetic study of the moduli spaces began with Jacobi's "Fundamenta Nova"