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Thesis

Analytic twists of modular forms and applications

We are interested in the study of non-correlation of Fourier coefficients of Maass forms against a wide class of real analytic functions. In particular, the class of functions we are interested in should be thought of as some archimedean analogs of Frobenius trace functions. In the first part of the thesis, we give an axiomatic definition for this class, and prove that these functions satisfy properties similar to that of Frobenius trace functions. In particular, we prove non-correlation statements analogous to those given by Fouvry, Kowalski and Michel for algebraic trace functions. In the second part of the thesis, we establish the existence of large values of Hecke-Maass L-functions with prescribed argument. In studying these problems, one encounters sums of Fourier coefficients of Maass forms against real oscillatory functions. In some cases, one can prove that these functions satisfy the axioms discussed previously.

    Keywords: Automorphic forms ; trace functions ; exponential sums ; L-functions

    Thèse École polytechnique fédérale de Lausanne EPFL, n° 7559 (2017)
    Programme doctoral Mathématiques
    Faculté des sciences de base
    Institut de mathématiques de géométrie et applications
    Chaire de théorie analytique des nombres
    Jury: Prof. Donna Testerman (présidente) ; Prof. Philippe Michel (directeur de thèse) ; Prof. Emmanuel Kowalski, Prof. Etienne Fouvry, Prof. Emmanuel Royer (rapporteurs)

    Public defense: 2017-6-1

    Reference

    Record created on 2017-05-24, modified on 2017-07-05

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