Foliage — Feuilletages et géométrie algébrique

Projet ANR-16-CE40-0008 de l'Agence nationale de la recherche (2016-2020)

Foliations over algebraic varieties defined over number fields: transcendence proofs and zero lemmas

Ile de Tatihou, 2–6 juillet 2018

Présentation

1) Arithmetic geometry for complex geometers: from fibrations to arithmetic schemes (2 hours; Carlo Gasbarri)

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.

2) Transcendence for algebraic and analytic geometers (6 hours; Jean-Benoît Bost)

I will give present some classical transcendence theorems in a way that will emphasize their interpretation as algebraization theorems concerning some combination of formal data over the integers and complex analytic data, and their analogy with some "geometric" algebraization theorems, such as Chow’s theorem or Lefschetz’ theorems à la Grauert-Grothendieck, concerning analytic or formal data.

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.

3) Zero estimates for solutions of differential equations (6 hours ; Gal Binyamini)

I will discuss the problem of estimating the order of zero of a polynomial restricted to the trajectory of a polynomial vector field. In particular I will present the topological approach of Gabrielov and the algebraic approach of Nesterenko, and some more recent variations. Depending on the interests of the audience I will also discuss some related questions, including:
- 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.

4) Recent developments and perspectives (4 hours)

The workshop will conclude with some talks (by Georges Comte and Tiago Jardim da Fonseca) devoted to some recent developments related to its main topics, and with a final talk (by the organizers) dedicated to research perspectives and open problems.

Bibliographie

  • 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.


Organisateurs

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