Estimates of Automorphic Functions

Abstract

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.

Multiple Dirichlet series and shifted convolutions

“…We prove a sharp bound for the average value of the triple product of modular functions for the Hecke subgroup Γ 0 (N ). Our result is an extension of the main result in [4] to a fixed cuspidal representation of the adele group PGL 2 (A).…”

“…The second is addressed by the following proposition, a generalization of [4], which is proved in the Appendix by Andre Reznikov (Theorem B): Proposition 4.1. With the notation as given above…”

“…We recall the setup of [4] which should be read in conjunction with this appendix. Let Y be a compact Riemann surface with a Riemannian metric of constant curvature −1 and the associated volume element dv.…”

“…In [4] we considered the following particular case of the triple period problem. Namely, we fix two Maass forms φ = φ τ , φ = φ τ as above and consider coefficients defined by the triple period:…”

“…We note first that one has exponential decay for the coefficients c i in the parameter |λ i | as i goes to ∞. For that reason, one renormalizes coefficients |c i | 2 by an appropriate ratio of Gamma functions dictated by the Watson formula [48] (see also Appendix in [4] where these factors were computed using another point of view). Taking into account the asymptotic behavior of these factors, we introduced normalized coefficients…”