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H 1 Symmetric power moments of Kloosterman sums

Auteurs : Fresán, Javier (Auteur de la Conférence)
CIRM (Editeur )

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    Résumé : We construct motives over the rational numbers associated with symmetric power moments of Kloosterman sums, and prove that their $L$-functions extend meromorphically to the complex plane and satisfy a functional equation conjectured by Broadhurst and Roberts. Although the motives in question turn out to be classical, the strategy consists in first realizing them as exponential motives and computing their Hodge numbers by means of the irregular Hodge filtration. We show that all Hodge numbers are either zero or one, which implies potential automorphicity thanks to recent results of Patrikis and Taylor. The first talk will be concerned with the arithmetic aspects and in the second one we will present the Hodge theoretic computations. Joint work with Claude Sabbah and Jeng-Daw Yu.

    Keywords : Kloosterman sums; power moments; exponential sums; Hodge filtration; De Rham cohomology

    Codes MSC :
    11L05 - Gauss and Kloosterman sums
    14C30 - Transcendental methods, Hodge theory, Hodge conjecture
    14F40 - de Rham cohomology
    32S40 - Monodromy; relations with differential equations and D-modules

      Informations sur la Vidéo

      Langue : Anglais
      Date de publication : 25/10/2018
      Date de captation : 15/10/2018
      Collection : Research talks ; Algebraic and Complex Geometry ; Number Theory
      Format : MP4
      Durée : 01:02:09
      Domaine : Algebraic & Complex Geometry ; Number Theory
      Audience : Chercheurs ; Doctorants , Post - Doctorants
      Download : https://videos.cirm-math.fr/2018-10-15_Fresan.mp4

    Informations sur la rencontre

    Nom de la rencontre : Cohomology of algebraic varieties / Cohomologie des variétés algébriques
    Organisateurs de la rencontre : Cadoret, Anna ; Charles, François ; Demarche, Cyril ; Klinger, Bruno ; Moonen, Ben
    Dates : 15/10/2018 - 19/10/2018
    Année de la rencontre : 2018
    URL Congrès : https://conferences.cirm-math.fr/1775.html

    Citation Data

    DOI : 10.24350/CIRM.V.19466903
    Cite this video as: Fresán, Javier (2018). Symmetric power moments of Kloosterman sums. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19466903
    URI : http://dx.doi.org/10.24350/CIRM.V.19466903


    Voir aussi

    Bibliographie

    1. Fresán, J., Sabbah, C., & Yu, J.-D. (2018). Hodge theory of Kloosterman connections. - https://arxiv.org/abs/1810.06454

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