# 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.

#### 12 Citations

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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). We… Expand

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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.

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The sup-norm problem for GL(2) over number fields

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- Israel Journal of Mathematics
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Let φ be a spherical Hecke–Maaß cusp form on the non-compact space PGL3(ℤ)PGL3(ℝ). We establish various pointwise upper bounds for φ in terms of its Laplace eigenvalue λφ. These imply, for φ… Expand

Explicit subconvexity savings for sup-norms of cusp forms on PGLn(R)

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Abstract Blomer and Maga [2] recently proved that, if F is an L 2 -normalized Hecke-Maass cusp form for SL n ( Z ) , and Ω is a compact subset of PGL n ( R ) / PO n ( R ) , then we have ‖ F | Ω ‖ ∞ ≪… Expand

Lower bounds for Maass forms on semisimple groups

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Let $G$ be an anisotropic semisimple group over a totally real number field $F$. Suppose that $G$ is compact at all but one infinite place $v_{0}$. In addition, suppose that $G_{v_{0}}$ is… Expand

Local analysis of Whittaker new vectors and global applications

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The main focus of this work is the supnorm problem for automorphic forms on GL2, and proves hybrid upper bounds, in other words estimates that are explicit in all major aspects of the Automorphic form under investigation. Expand

#### References

SHOWING 1-10 OF 34 REFERENCES

The Sup-Norm Problem for PGL(4)

- Mathematics
- 2014

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). We… Expand

Restrictions of SL_3 Maass forms to maximal flat subspaces

- Mathematics
- 2013

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

- Mathematics
- 2014

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 coset… Expand

Bounds for eigenforms on arithmetic hyperbolic $3$-manifolds

- Mathematics
- 2016

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 bound… Expand

Geodesic restrictions of arithmetic eigenfunctions

- Mathematics
- 2012

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 \psi… Expand

Eisenstein Series First Part

- Mathematics
- 2005

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 universal… Expand

$L^p$ norms of higher rank eigenfunctions and bounds for spherical functions

- Mathematics
- 2016

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 to… Expand

Posn(r) and Eisenstein Series

- Mathematics
- 2005

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.- Eisenstein… Expand

Automorphic Forms and L-Functions for the Group Gl(n, R)

- Mathematics
- 2006

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 for… Expand

Holomorphic extensions of representations: (I) automorphic functions

- Mathematics
- 2002

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 that… Expand