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H 1 Understanding quadratic forms on lattices through generalised theta series

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

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 calculsOrganisateurs de la rencontre : Bayer-Fluckiger, Eva ; Elbaz-Vincent, Philippe ; Ellis, Graham ; Gunnels, PaulDates : 25/03/2019 - 29/03/2019 Année de la rencontre : 2019 URL Congrès : https://conferences.cirm-math.fr/1995.htmlCitation 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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