Valeurs centrales et valeurs au bord de la bande critique de fonctions L automorphes

Abstract : Special values of automorphic L-functions are considered in this work in three parts. In the first part, elementary information about automorphic forms and as- sociated symmetic power L-functions, which will be very useful in the following parts, is introduced. In the second part, we study the central values, in the form of higher mo- ment in short interval, of automorphic L-functions and give a proof for the conjecture of Conrey et al. to get the sharp bound for the moment under Generalized Riemann Hypothesis. In the last part, values of automorphic L-functions at s = 1 are conside- red in level-weight aspect. We generalize and/or improve related early results about the bounds of values at s = 1, the Montgomery-Vaughan’s conjecture and distribu- tion functions. As an application of our results on extreme values, the distribution of coefficients of newforms concerning the Sato-Tate conjecture is studied.
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Xuanxuan Xiao. Valeurs centrales et valeurs au bord de la bande critique de fonctions L automorphes. Théorie des nombres [math.NT]. Université de Lorraine, 2015. Français. ⟨NNT : 2015LORR0068⟩. ⟨tel-01751699v2⟩

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