06/24/2008
Laboratoire de mathématiques Nicolas Oresme
CNRS UMR 6139
Université de Caen BP 5186
14032 Caen cedex, France.
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Location: Campus 2 Building: Sciences 3 Room: 108 |
Tel.: (+33) 2 31 56 74 23 Fax : (+33) 2 31 56 73 20 E-mail: pontreau(a)math.unicaen.fr |
Since September 2005, I am a Teaching and Research Associate (ATER) at the University of Caen in the laboratoire de mathématiques Nicolas Oresme.
My curriculum vitae [pdf].
Title: Lower bounds for the normalized height in small codimension
Obtained: December 09 2005
Examiners:
Complete text (in French): [pdf] and [dvi].
définie sur 
, Acta Arith. 120 (1), pp. 1-26, (2005).
Abstract. We are concerned here with Lehmer's problem in dimension 2; we give a lower bound for the height of a non-torsion point of on a non-torsion curve defined over , depending on the degree of the curve only. We have first been inspired by an article of F. Amoroso and S. David (see Distribution des points de petite hauteur dans les groupes multiplicatifs Ann. Scuola Norm. Sup. Pisa Sci. Serie V Vol III Fasc. 2 (2004)); we develop a new approach, inherent in the dimension two (or more precisely the codimension two), and then obtain a better result where the error's term is improved significantly, moreover we give an explicit expression for the constant.
Abstract. This article deals with Bogomolov's problem for the hypersurfaces; we give a
geometric lower bound for the height of a hypersurface of 
(i.e. without condition on the field of definition of the hypersurface) which is not a translate of an algebraic
subgroup of . This is an analogue of a result of F. Amoroso and S. David who give a lower bound for the height of non-torsion hypersurfaces defined and irreducible over the rationals.
Abstract. For any geometrically irreducible variety V of the multiplicative group , one knows that outside finitely number of exceptional translated subtori included in V, all the points have a height bounded by below by some quantity q(V)-1>0. Moreover one can give an upper bound for the sum of the degrees of these translated subtori, when q(V) is polynomial in the degree of V. This is not the case if we want a quasi-optimal lower bound for the height of the points of V, essentially linear in the inverse of the degree.
We give here a partial answer to this problem: we give an upper bound for the sum of the degrees of these exceptional translated subtori of codimension 1 of a hypersurface. The results, obtained in the case of 
, are completely explicit, and could be extended to , with some slight complications inherent to dimension n.
, accepted for publication in Monatshefte für Mathematik(2008).Abstract. We prove a sharper so-called Mordell-Lang plus Bogomolov type result for curves lying in the two-dimensional linear torus. We mainly follow the approach of G. Rémond (see Sur les sous-variétés des tores, Compositio Mathematica, 2002 Vol. 134, N. 3), using Vojta and Mumford inequalities. In the special case we consider, we improve Rémond's main result using a better Bogomolov property and an elementary arithmetic Bézout theorem.
Abstract. This notes are an introduction to Chow's forms and normalized heights in the tori, following presentation of Patrice Philippon and Sinnou David (see P. Philippon, Critères pour l'indépendance algébrique, Inst. Hautes Etudes Sci. Publ. Math., 1986, 64, pp. 5-52 and S. David and P. Philippon, Minorations des hauteurs normalisées des sous-variétés des tores, Scuola Norm. Sup. Sci., 1999, 28 (4), pp. 489-543). In view of a self-contained document, we give in a first part somes commutative algebra's resultats, used forward, concerning in particular notions of dimension, primary decompositions, and Hilbert's polynomial.