# ESTIMATES OF AUTOMORPHIC FUNCTIONS

@article{Bernstein2003ESTIMATESOA, title={ESTIMATES OF AUTOMORPHIC FUNCTIONS}, author={Joseph Bernstein and Andre Reznikov}, journal={Moscow Mathematical Journal}, year={2003}, volume={4}, pages={19-37} }

We present a new method to estimate trilinear period for automorphic representations of SL2(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 gen- eral method which gives an estimate for the average of the automorphic factor and thus prove a convexity bound for the triple period. 2000 Math. Subj. Class… Expand

#### 52 Citations

Periods, Subconvexity of L-functions and Representation Theory

- Mathematics
- 2005

We describe a new method to estimate the trilinear period on automorphic representations of PG L2(R). Such a period gives rise to a special value of the triple L-function. We prove a bound for the… Expand

Subconvexity bounds for triple L-functions and representation theory

- Mathematics
- 2006

We describe a new method to estimate the trilinear period on automorphic representations of PGL 2 (ℝ). Such a period gives rise to a special value of the triple L-function. We prove a bound for the… Expand

Automorphic Periods and Representation Theory ( from the analytic perspective )

- 2011

In these notes, we discuss periods of automorphic functions from the analytic perspective. We consider an effective version of the Frobenius reciprocity of Gelfand and Fomin. It turns out that it is… Expand

Estimates of triple products of automorphic functions II

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- 2012

We prove a sharp bound for the average value of the triple product of modular functions for the Hecke subgroup \Gamma_0(N). Our result is an extension of the main result in {Bernstein&Reznikov-2004}… Expand

DEPENDENCY ON THE GROUP IN AUTOMORPHIC SOBOLEV INEQUALITIES

- 2008

In [1] and [2], Bernstein and Reznikov have introduced a new way of estimating the coefficients in the spectral expansion of φ2, where φ is a Maass cusp of norm 1 on a quotient Y = Γ\H of the… Expand

Rankin-Selberg without unfolding and bounds for spherical Fourier coefficients of Maass forms

- Mathematics
- 2005

In this paper we study periods of automorphic functions. We present a new method which allows one to obtain non-trivial spectral identities for weighted sums of certain periods of automorphic… Expand

Lattice points counting and bounds on periods of Maass forms

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- Transactions of the American Mathematical Society
- 2019

We provide a "soft" proof for non-trivial bounds on spherical, hyperbolic and unipotent Fourier coefficients of a fixed Maass form for a general co-finite lattice $\Gamma$ in $PGL(2,R)$. We use the… Expand

A uniform bound for geodesic periods of eigenfunctions on hyperbolic surfaces

- Mathematics
- 2015

We consider periods along closed geodesics and along geodesic circles for eigenfunctions of the Laplace-Beltrami operator on a compact hyperbolic Riemann sur- face. We obtain uniform bounds for such… Expand

Microlocal lifts of eigenfunctions on hyperbolic surfaces and trilinear invariant functionals

- Mathematics
- 2004

S. Zelditch introduced an equivariant version of a pseudo-differential calculus on a hyperbolic Riemann surface. We recast his construction in terms of trilinear invariant functionals on irreducible… Expand

Upper bounds for geodesic periods over rank one locally symmetric spaces

- Mathematics
- Forum Mathematicum
- 2018

We prove upper bounds for geodesic periods of automorphic forms over general rank one locally symmetric spaces. Such periods are integrals of automorphic forms restricted to special totally geodesic… Expand

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