Whittaker coefficients of automorphic forms and applications to analytic Number Theory
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The formalism of automorphic representations makes the study of automorphic forms amenable to representation-theoretic methods. In particular the Whittaker model, when it exists, permits to extract interesting arithmetic and analytic information. In this thesis, we give two instances of this principle, in which we are concerned respectively with a) bounding the values taken by and b) the distribution of the Satake parameters of certain automorphic forms. In the first part of this thesis, carried out in Chapter 1, we study the problem of bounding the sup norms of L 2 -normalized cuspidal automorphic newforms ϕ on GL2 in the level aspect. Prior to this work, strong upper bounds were only available if the central character χ of ϕ is not too highly ramified. We establish a uniform upper bound in the level aspect for general χ. If the level N is a square, our result reduces to ∥ϕ∥∞ ≪ N 1 4 +ϵ , at least under the Ramanujan Conjecture. In particular, when χ has conductor N, this improves upon the previous best known bound ∥ϕ∥∞ ≪ N 1 2 +ϵ in this setup (due to Saha) and matches a lower bound due to Templier, thus our result is essentially optimal in this case. In the second and more substantial part, carried out in Chapter 2, we develop a Kuznetsov type formula for the group GSp4 . To this end, we follow a relative trace formula approach, and we focus on giving a final formula that is as explicit as possible. In particular, our formula is valid for arbitrary level, arbitrary central character, and includes the Hecke eigenvalues. We then use this Kuznetsov formula in Chapter 3 to show that, as the level tends to infinity, the Satake parameters of automorphic forms on GSp4 , suitably weighted, equidistribute with respect to the Sato-Tate measure.
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Comtat, FCollections
- Theses [4164]