Hybrid bounds for automorphic forms on ellipsoids over number fields

Blomer, Valentin; Michel, Philippe

doi:10.1017/s1474748012000874

Cited by 17 publications

(24 citation statements)

References 17 publications

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“…For L 2 -normalized Hecke Maaß cusp forms F they proved the bound F ∞ ≪ (1 + λ F ) 5/24+ε . Similar results have been obtained for the Hecke-Laplace eigenfunctions on the sphere and ellipsoids [Van97,BM13] and on congruence quotients of hyperbolic 3-space [BHM]. The underlying algebraic groups in these cases are SL 2 (R) = SO 0 (2, 1), SO(3) and SL 2 (C) = SO 0 (3, 1), all of which have real rank at most 1.…”

Section: Introductionsupporting

confidence: 77%

The sup-norm problem on the Siegel modular space of rank two

Blomer¹,

Pohl²

2016

American Journal of Mathematics

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Abstract. Let F be a square integrable Maaß form on the Siegel upper half space H of rank 2 for the Siegel modular group Sp 4 (Z) with Laplace eigenvalue λ. If, in addition, F is a joint eigenfunction of the Hecke algebra and Ω is a compact set in Sp 4 (Z)\H, we show the bound F | Ω ∞ ≪ Ω (1 + λ) 1−δ for some global constant δ > 0. As an auxiliary result of independent interest we prove new uniform bounds for spherical functions on semisimple Lie groups.

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Section: Introductionsupporting

confidence: 77%

The sup-norm problem on the Siegel modular space of rank two

Blomer¹,

Pohl²

2016

American Journal of Mathematics

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“…The following result is an essentially sharp refinement of this count that takes into account the finer shape of Λ. It is a kind of Lipschitz principle, also implicit in the corresponding diophantine analysis for the sup-norm problem on SO 3 (R) [BM,Section 4].…”

Section: Geometry Of Numbersmentioning

confidence: 98%

Bounds for eigenforms on arithmetic hyperbolic 3-manifolds

Blomer¹,

Harcos²,

Milićević³

2016

Duke Math. J.

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On a family of arithmetic hyperbolic 3-manifolds of squarefree level, we prove an upper bound for the sup-norm of Hecke-Maaß cusp forms, with a power saving over the local geometric bound simultaneously in the Laplacian eigenvalue and the volume. By a novel combination of diophantine and geometric arguments in a noncommutative setting, we obtain bounds as strong as the best corresponding results on arithmetic surfaces.2000 Mathematics Subject Classification. Primary 11F72, 11F55, 11J25.1 Very recently, a remarkable non-arithmetic approach for classical holomorphic cusp forms was given in [FJK]. 2 Unfortunately, the argument in this work seems to have a gap, specifically the proof of [Ko, Lemma 5.3] is incomplete.Our counting argument is quite different from Koyama's method, though at one point we incorporate a crucial idea from his paper (see Section 11.2).

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“…σ 0 ( \G) ∼ = L 2 ( \G/K ) ∼ = L 2 (S 2 ) is treated, S 2 being the 2-sphere. In the papers [6,7], hybrid L ∞ -norms for general arithmetic quotients of 2-spheres in the eigenvalue and level aspect are studied. However, the exponents for the spectral parameter are a little greater than 5/24 there, namely 1…”

Section: Subconvex Bounds On So(3)mentioning

confidence: 99%

“…the mentioned assumptions must hold by Lemma 5.1. This choice is essentially the same as in the work of Blomer-Maga [6,7] on SL(n, Z) ⊂ PGL(n, R) in the case n = 2.…”

Section: Remark 72mentioning

confidence: 99%

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Subconvex bounds for Hecke–Maass forms on compact arithmetic quotients of semisimple Lie groups

Ramacher

Wakatsuki

2020

Math. Z.

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Let H be a semisimple algebraic group, K a maximal compact subgroup of $$G:=H({{\mathbb {R}}})$$ G : = H ( R ) , and $$\Gamma \subset H({{\mathbb {Q}}})$$ Γ ⊂ H ( Q ) a congruence arithmetic subgroup. In this paper, we generalize existing subconvex bounds for Hecke–Maass forms on the locally symmetric space $$\Gamma \backslash G/K$$ Γ \ G / K to corresponding bounds on the arithmetic quotient $$\Gamma \backslash G$$ Γ \ G for cocompact lattices using the spectral function of an elliptic operator. The bounds obtained extend known subconvex bounds for automorphic forms to non-trivial K-types, yielding such bounds for new classes of automorphic representations. They constitute subconvex bounds for eigenfunctions on compact manifolds with both positive and negative sectional curvature. We also obtain new subconvex bounds for holomorphic modular forms in the weight aspect.

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Hybrid bounds for automorphic forms on ellipsoids over number fields

Cited by 17 publications

References 17 publications

The sup-norm problem on the Siegel modular space of rank two

The sup-norm problem on the Siegel modular space of rank two

Bounds for eigenforms on arithmetic hyperbolic 3-manifolds

Subconvex bounds for Hecke–Maass forms on compact arithmetic quotients of semisimple Lie groups

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