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Comptes Rendus Mathématique
Volume 351, n° 13-14
pages 523-526 (juillet 2013)
Doi : 10.1016/j.crma.2013.05.003
Received : 16 October 2012 ;  accepted : 15 May 2013
Dérivation relative
Differentiating relatively
 

Michael McQuillan
 Dipartimento di Matematica, Università degli studi di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Roma, Italy 

Résumé

Étant donnée une suite   de points algébriques dʼune variété X sur un corps de fonctions de caractéristique zéro, K , avec une hauteur (normalisée) tendant vers lʼinfini, nous construisons une dérivée de la suite dans  . La vraie (Oesterlé, 2002 [[4]]) conjecture « a, b, c » sur les corps des fonctions est un corollaire immédiat. En principe, chaque problème du type Mordell sur les corps des fonctions se réduit, par cette construction, au problème hyperbolique correspondant sur la fibre générique, mais, malheureusement, une telle conclusion est délicate en présence de mauvaise réduction. Les résultats présentés ici – comme on les trouve dans McQuillan (2001) [[3]] – ainsi quʼune autre approche de la conjecture « a, b, c » par K. Yamanoi (2004) [[5]] ont été déjà reportés dans le cadre du séminaire Bourbaki (Gasbarri, 2008 [[1]]).

The full text of this article is available in PDF format.
Abstract

Given a sequence of algebraic points   of a variety X over a characteristic 0-function field K of unbounded (normalised) height, we construct a limiting derivative in  . The real (Oesterlé, 2002 [[4]]) “a, b, c” conjecture over function fields is an immediate corollary. In principle, every Mordellic problem over function fields reduces to a hyperbolicity problem on the generic fibre by way of the said construction, but, unfortunately, such a conclusion is delicate in the presence of bad reduction. This – as found in McQuillan (2001) [[3]] – together with an alternative approach to the “a, b, c” conjecture by K. Yamanoi (2004) [[5]] has already been reported in the Séminaire Bourbaki (Gasbarri, 2008 [[1]]).

The full text of this article is available in PDF format.


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