On the sup-norm of SL 3 Hecke–Maass cusp forms

@article{Holowinsky2014OnTS,
  title={On the sup-norm of SL 3 Hecke–Maass cusp forms},
  author={Roman Holowinsky and K. Nowland G. Ricotta and Emmanuel Royer},
  journal={arXiv: Number Theory},
  year={2014}
}
This work contains a proof of a non-trivial explicit quantitative bound in the eigenvalue aspect for the sup-norm of a SL(3,Z) Hecke-Maass cusp form restricted to a compact set. 

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References

SHOWING 1-10 OF 34 REFERENCES
The Sup-Norm Problem for PGL(4)
Let F be a Hecke-Maass cusp form for the group SL(4, Z) with Laplace eigenvalue lambda. Assume that F satisfies the Ramanujan conjecture at infinity (this is satisfied by almost all cusp forms). WeExpand
Restrictions of SL_3 Maass forms to maximal flat subspaces
Let \psi be a Hecke-Maass form on a cubic division algebra over \Q. We apply arithmetic amplification to improve the local bound for the L^2 norm of \psi restricted to maximal flat subspaces.
The amplification method in the GL(3) Hecke algebra
This article contains all of the technical ingredients required to implement an effective, explicit and unconditional amplifier in the context of GL(3) automorphic forms. In particular, several cosetExpand
Bounds for eigenforms on arithmetic hyperbolic $3$-manifolds
On a family of arithmetic hyperbolic 3-manifolds of square-free level, we prove an upper bound for the sup-norm of Hecke–Maas cusp forms, with a power saving over the local geometric boundExpand
Geodesic restrictions of arithmetic eigenfunctions
Let X be an arithmetic hyperbolic surface, \psi a Hecke-Maass form, and l a geodesic segment on X. We obtain a power saving over the local bound of Burq-G\'erard-Tzvetkov for the L^2 norm of \psiExpand
Eisenstein Series First Part
In this chapter, we start systematically to investigate what happens when we take the trace over the discrete groups γ = GLn(Z), for various objects. In the first section, we describe a universalExpand
$L^p$ norms of higher rank eigenfunctions and bounds for spherical functions
We prove almost sharp upper bounds for the $L^p$ norms of eigenfunctions of the full ring of invariant differential operators on a compact locally symmetric space, as well as their restrictions toExpand
Posn(r) and Eisenstein Series
GLn (R) actions on Posn(R).- Measures, Integration, and Quadratic Model.- Special Functions on Posn(R).- Invariant Differential Operators on Posn(R).- Poisson duality and zeta functions.- EisensteinExpand
Automorphic Forms and L-Functions for the Group Gl(n, R)
Introduction 1. Discrete group actions 2. Invariant differential operators 3. Automorphic forms and L-functions for SL(2,Z) 4. Existence of Maass forms 5. Maass forms and Whittaker functions forExpand
Holomorphic extensions of representations: (I) automorphic functions
Let G be a connected, real, semisimple Lie group contained in its complexification GC, and let K be a maximal compact subgroup of G. We construct a KC-G double coset domain in GC, and we show thatExpand
...
1
2
3
4
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