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H 1 Shimura curves and bounds for the $abc$ conjecture

Auteurs : Pasten, Hector (Auteur de la Conférence)
CIRM (Editeur )

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    Résumé : I will explain some new connections between the $abc$ conjecture and modular forms. In particular, I will outline a proof of a new unconditional estimate for the $abc$ conjecture, which lies beyond the existing techniques in this context. The proof involves a number of tools such as Shimura curves, CM points, analytic number theory, and Arakelov geometry. It also requires some intermediate results of independent interest, such as bounds for the Manin constant beyond the semi-stable case. If time permits, I will also explain some results towards Szpiro's conjecture over totally real number fields which are compatible with the discriminant term appearing in Vojta's conjecture for algebraic points of bounded degree.

    Codes MSC :
    11F11 - Holomorphic modular forms of integral weight
    11G05 - Elliptic curves over global fields
    11G18 - Arithmetic aspects of modular and Shimura varieties
    14G40 - Arithmetic varieties and schemes; Arakelov theory; heights

      Informations sur la Vidéo

      Langue : Anglais
      Date de publication : 29/05/2018
      Date de captation : 23/05/2018
      Collection : Research talks ; Algebraic and Complex Geometry ; Number Theory
      Format : MP4
      Durée : 01:00:35
      Domaine : Algebraic & Complex Geometry ; Number Theory
      Audience : Chercheurs ; Doctorants , Post - Doctorants
      Download : https://videos.cirm-math.fr/2018-05-23_Pasten.mp4

    Informations sur la rencontre

    Nom de la rencontre : Diophantine geometry / ​Géométrie diophantienne
    Organisateurs de la rencontre : Bosser, Vincent ; Carrizosa, Maria ; Gaudron, Eric ; Habegger, Philipp
    Dates : 21/05/2018 - 25/05/2018
    Année de la rencontre : 2018
    URL Congrès : https://conferences.cirm-math.fr/1754.html

    Citation Data

    DOI : 10.24350/CIRM.V.19408103
    Cite this video as: Pasten, Hector (2018). Shimura curves and bounds for the $abc$ conjecture. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19408103
    URI : http://dx.doi.org/10.24350/CIRM.V.19408103

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