Intersections improbables effectives

par Emanuele Tron

Projet de thèse en Mathématiques Pures

Sous la direction de Yuri Bilu.

Thèses en préparation à Bordeaux , dans le cadre de École doctorale de mathématiques et informatique , en partenariat avec IMB - Institut de Mathématiques de Bordeaux (laboratoire) et de Théorie des nombres (equipe de recherche) depuis le 11-09-2018 .


  • Résumé

    The theory of Unlikely Intersections, pioneered in 90ies in the work of Bombieri, Masser and Zannier, became now one of the leading research fields in the modern Arithmetic Geometry. Some spectacular results like partial proofs of Zilber-Pink conjecture and the André-Oort conjecture are obtained in the work of celebrated mathematicians like André, Edixhoven, Habegger, Masser, Pila, Rémond, Viada, Yafaev, Zannier and many other younger researchers. One may mention Pila's plenary talk on the ICM 2014 entirely dedicated to this topic. Unfortunately, many of these results are either conditional on GRH or non-effective; that is, do not imply any algorithm for actual solving the problem. This is especially true for the results related to the André-Oort conjecture. The general trend of modern Diophantine Geometry is making the things as effective as possible. The further trend is making producing for effective results working algorithms to make them explicit. One recent example is complete classification of special points on straight lines by Allobert, Bilu and Pizarro. Effective Unlikely Intersections is a hot topic now. For instance, a conference dedicated specifically to this topic will be held in Manchester in July 2018. In this project we plan to obtain effective unconditional proofs of many results of André-Oort type, and make some of them completely explicit.

  • Titre traduit

    Effective Unlikely Intersections


  • Résumé

    The theory of Unlikely Intersections, pioneered in 90ies in the work of Bombieri, Masser and Zannier, became now one of the leading research fields in the modern Arithmetic Geometry. Some spectacular results like partial proofs of Zilber-Pink conjecture and the André-Oort conjecture are obtained in the work of celebrated mathematicians like André, Edixhoven, Habegger, Masser, Pila, Rémond, Viada, Yafaev, Zannier and many other younger researchers. One may mention Pila's plenary talk on the ICM 2014 entirely dedicated to this topic. Unfortunately, many of these results are either conditional on GRH or non-effective; that is, do not imply any algorithm for actual solving the problem. This is especially true for the results related to the André-Oort conjecture. The general trend of modern Diophantine Geometry is making the things as effective as possible. The further trend is making producing for effective results working algorithms to make them explicit. One recent example is complete classification of special points on straight lines by Allobert, Bilu and Pizarro. Effective Unlikely Intersections is a hot topic now. For instance, a conference dedicated specifically to this topic will be held in Manchester in July 2018. In this project we plan to obtain effective unconditional proofs of many results of André-Oort type, and make some of them completely explicit.