These lectures will present some basic notions of Diophantine geometry — such as models over the integers of quasi-projective varieties over a number field, hermitian vector bundles, Arakelov degree, or heights — that play a central role in transcendence proofs, in a manner that will emphasize the analogy with the geometric notions of varieties fibered over over curve, of vector bundles, and of degree of a projective variety with respect to some line bundle, familiar to complex geometers.

In the first part of the lectures, I will focus on the theorem of Schneider-Lang, and present a proof of this theorem formally parallel to some simple proof of Chow’s theorem. In a second part, I will discuss some transcendence results — such as Baker’s results on "linear forms in logarithms" and the theorem of Siegel-Shidlovski — the proofs of which crucially involve some zero lemma.

Foliations on algebraic varieties over number fields and the algebraicity properties of their leaves will be a unifying theme behind the various transcendence results I will discuss.

- The relation between zero estimates and a problem in differential algebra; and applications for the geometry of transcendental numbers (following Pillay and Hrushovski).

- The generalization of zero estimates to foliations of dimension greater than one (following Gabrielov-Khovanskii and other more recent results).

- Applications of zero estimates around the Pila-Wilkie theorem and Wilkie conjecture on counting rational points in transcendental sets.

- G. Binyamini and D. Novikov : The Pila-Wilkie theorem for subanalytic families:
a complex analytic approach.
*Compos. Math.***153**(2017) 2171–2194. - G. Binyamini : Multiplicity estimates: a Morse-theoretic approach.
*Duke Math. J.***165**(2016) 95–128. - J.-B. Bost : Algebraic leaves of algebraic foliations over number fields.
*Publ. Math. I.H.E.S.***93**(2001), 161-221. - J.-B. Bost : Algebraization, transcendence, and D-group schemes.
*Notre Dame J. Form. Log.***54**(2013) 377–434. - G. Comte and Y. Yomdin : Zeroes and rational points of analytic functions.
To appear in
*Ann. Inst. Fourier (Grenoble)*. - S. Fischler, É. Gaudron and S. Khémira (eds.) Formes modulaires et transcendance.
Colloque "jeunes".
*Séminaires et Congrès***12**(2005) 271 pp. - T. J. Fonseca : Higher Ramanujan equations I: moduli stacks of abelian varieties and higher Ramanujan vector fields. arXiv:1612.05081
- T. J. Fonseca : Higher Ramanujan equations II: periods of abelian varieties and transcendence questions. arXiv:1703.02954
- T. J. Fonseca : Algebraic independence for values of integral curves. arXiv:1710.00563
- C. Gasbarri : Analytic subvarieties with many rational points.
*Math. Ann.***346**(2010) 199–243. - C. Gasbarri : Horizontal sections of connections on curves and transcendence.
*Acta Arith.***158**(2013) 99–128. - A. Gabrielov : Multiplicities of zeroes of polynomials on trajectories of polynomial vector fields and bounds on degree of nonholonomy.
*Math. Res. Lett.***2**(1995) 437–451. - A. Gabrielov :
Multiplicity of a zero of an analytic function on a trajectory of a vector field.
The Arnoldfest (Toronto, ON, 1997) 191-200,
*Fields Inst. Commun.***24**Amer. Math. Soc., Providence, RI, 1999. - D. Masser, Yu. V. Nesterenko, H. P. Schlickewei, W. M. Schmidt and M. Waldschmidt :
Diophantine approximation. Lectures from the C.I.M.E. Summer School held in Cetraro, June 28–July 6, 2000. Edited by F. Amoroso and U. Zannier
*Lecture Notes in Mathematics***1819**Springer-Verlag, Berlin. - Yu. V. Nesterenko : Modular functions and transcendence questions.
*Mat. Sb.***187**(1996) 65-96; translation in*Sb. Math.***187**(1996) 1319–1348. - Yu. V. Nesterenko : Estimates for the number of zeros of certain functions.
in New Advances in Transcendence Theory (Durham, 1986), Cambridge Univ. Press, Cambridge, 1988,
Nesterenko and Philippon (eds), Introduction to algebraic independence theory.
*Lecture Notes in Mathematics***1752**. Springer-Verlag, Berlin, 2001.

Jean-Benoît Bost, Carlo Gasbarri, Frank Loray, Erwan Rousseau