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H 1 Understanding quadratic forms on lattices through generalised theta series

Auteurs : Walling, Lynne (Auteur de la Conférence)
CIRM (Editeur )

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    Résumé : Siegel introduced generalised theta series to study representation numbers of quadratic forms. Given an integral lattice $L$ with quadratic form $q$, Siegel’s degree $n$ theta series attached to $L$ has a Fourier expansion supported on $n$-dimensional lattices, with Fourier coefficients that tells us how many times $L$ represents any given $n$-dimensional lattice. Siegel proved that this theta series is a type of automorphic form.
    In this talk we explore how the theory of automorphic forms, together with the theory of quadratic forms, helps us understand these representation numbers. We reveal arithmetic relations between ”average” representation numbers (where we average over a genus), and finally we give an explicit formula for these average representation numbers in terms of the Fourier coefficients of Siegel Eisenstein series. In the case that $n = 1$ (meaning we are looking at how often $L$ represents an integer) this yields explicit numerical formulas for these average representation numbers.

    Keywords : Siegel theta series; representation number of indefinite quadratic forms; Eisenstein series; nonanalytic modular form; quadratic forms; average representation numbers

    Codes MSC :
    11F27 - Theta series; Weil representation; theta correspondences
    11F30 - Fourier coefficients of automorphic forms
    11F46 - Siegel modular groups and their modular and automorphic forms

      Informations sur la Vidéo

      Langue : Anglais
      Date de publication : 17/04/2019
      Date de captation : 28/03/2019
      Collection : Research talks ; Number Theory
      Format : MP4
      Durée : 00:45:07
      Domaine : Number Theory
      Audience : Chercheurs ; Doctorants , Post - Doctorants
      Download : https://videos.cirm-math.fr/2019-03-28_Walling.mp4

    Informations sur la rencontre

    Nom de la rencontre : Cohomology of arithmetic groups, lattices and number theory: geometric and computational viewpoint / Cohomologie des groupes arithmétiques, réseaux et théorie des nombres: géométries et calculs
    Organisateurs de la rencontre : Bayer-Fluckiger, Eva ; Elbaz-Vincent, Philippe ; Ellis, Graham ; Gunnels, Paul
    Dates : 25/03/2019 - 29/03/2019
    Année de la rencontre : 2019
    URL Congrès : https://conferences.cirm-math.fr/1995.html

    Citation Data

    DOI : 10.24350/CIRM.V.19509103
    Cite this video as: Walling, Lynne (2019). Understanding quadratic forms on lattices through generalised theta series. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19509103
    URI : http://dx.doi.org/10.24350/CIRM.V.19509103


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