Bounding hyperbolic and spherical coefficients of Maass forms

@article{Blomer2014BoundingHA,
  title={Bounding hyperbolic and spherical coefficients of Maass forms},
  author={Valentin Blomer and Farrell Brumley and Alex Kontorovich and Nicolas Templier},
  journal={Journal de Theorie des Nombres de Bordeaux},
  year={2014},
  volume={26},
  pages={559-578}
}
We develop a new method to bound the hyperbolic and spherical Fourier coefficients of Maass forms defined with respect to arbitrary uniform lattices. Let Σ be a compact hyperbolic surface, endowed with a metric of constant negative curvature −1. Let f be a non-constant eigenfunction of the Laplacian (∆ + λ)f = 0 on Σ, normalized so that ||f ||2 = 1. Let C be a closed geodesic or a geodesic circle in Σ. (By a geodesic circle we mean the set of points of fixed positive distance from a given point… 

Lattice points counting and bounds on periods of Maass forms

  • A. ReznikovF. Su
  • Mathematics
    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

Effective equidistribution of shears and applications

A “shear” is a unipotent translate of a cuspidal geodesic ray in the quotient of the hyperbolic plane by a non-uniform discrete subgroup of $${\text {PSL}}(2,\mathbb {R})$$PSL(2,R), possibly of

Effective equidistribution of shears and applications

A “shear” is a unipotent translate of a cuspidal geodesic ray in the quotient of the hyperbolic plane by a non-uniform discrete subgroup of PSL(2,R)\documentclass[12pt]{minimal} \usepackage{amsmath}

References

SHOWING 1-10 OF 22 REFERENCES

Limits of translates of divergent geodesics and integral points on one-sheeted hyperboloids

For any non-uniform lattice Γ in SL2(ℝ), we describe the limit distribution of orthogonal translates of a divergent geodesic in Γ\SL2(ℝ). As an application, for a quadratic form Q of signature (2,

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

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

Density of integer points on affine homogeneous varieties

(1.2) N(T, V)= {m V(Z): Ilmll T} where we denote by V(A), for any ring A, the set of A-points of V. Hence I1" is some Euclidean norm on R". The only general method available for such problems is the

Geodesic restrictions for the Casimir operator

Subconvexity bounds for triple L-functions and representation theory

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

Sparse equidistribution problems, period bounds and subconvexity

We introduce a "geometric" method to bound periods of automorphic forms. The key features of this method are the use of equidistribution results in place of mean value theorems, and the systematic

Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces

1. Ergodic systems 2. The geodesic flow of Riemannian locally symmetric spaces 3. The vanishing theorem of Howe and Moore 4. The horocycle flow 5. Siegel sets, Mahler's criterion and Margulis' lemma

Automorphic Forms and Representations

1. Modular forms 2. Automorphic forms and representations of GL( 2, R) 3. Automorphic representations 4. GL(2) over a p-adic field.

Fourth moments of grossencharakteren zeta functions