We present a new method of estimating trilinear period for automorphic representations of SL 2 (R). The method is based on the uniqueness principle in representation theory. We show how to separate the exponentially decaying factor in the triple period from the essential automorphic factor which behaves polynomially. We also describe a general method which gives an estimate on the average of the automorphic factor and thus prove a convexity bound for the triple period.
We describe a new method to estimate the trilinear period on automorphic representations of PGL 2 ./ޒ. Such a period gives rise to a special value of the triple Lfunction. We prove a bound for the triple period which amounts to a subconvexity bound for the corresponding special value of the triple L-function. Our method is based on the study of the analytic structure of the corresponding unique trilinear functional on unitary representations of PGL 2 ./ޒ.
ABSTRACT. This paper deals with some questions that have received a lot of attention since they were raised by Hejhal and Rackner in their 1992 numerical computations of Maass forms. We establish sharp upper and lower bounds for the L 2 -restrictions of these forms to certain curves on the modular surface. These results, together with the Lindelof Hypothesis and known subconvex L ∞ -bounds are applied to prove that locally the number of nodal domains of such a form goes to infinity with its eigenvalue.
Dedicated to Winnie Li with admirationKeywords: Subconvexity bounds Second moments Shifted convolutions L-functions of automorphic formsWe define, and obtain the meromorphic continuation of, shifted Rankin-Selberg convolutions in one and two variables. As sample applications, this continuation is used to obtain estimates for single and double shifted sums and a Burgesstype bound for L-series associated to modular forms of arbitrary central character. Further applications are furnished by subsequent works by the authors and their colleagues.
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