The sup-norm problem beyond the newform

Assing, Edgar

doi:10.48550/arxiv.2111.01893

Cited by 2 publications

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“…The estimate (1.5) is comparable in strength to the Weyl bound for the Riemann zeta function and has long been regarded as a natural limit for the sup-norm problem in the squarefree level aspect [HT13,Remarks (i)]. It has been extended to number fields [BHM16,BHMM20,Ass24] and to more general vectors than newforms [HNS19,Ass21]. For levels that are not squarefree (e.g., powers of a fixed prime), the flavor of the problem is quite different (see Remark 1.4), and stronger estimates have been achieved in [Sah17,Mar16,Sah20,Com21,HS20].…”

Section: Introductionmentioning

confidence: 94%

Theta functions, fourth moments of eigenforms and the sup-norm problem II

Khayutin,

Nelson,

Steiner

2024

Forum of Mathematics, Pi

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Let f be an $L^2$ -normalized holomorphic newform of weight k on $\Gamma _0(N) \backslash \mathbb {H}$ with N squarefree or, more generally, on any hyperbolic surface $\Gamma \backslash \mathbb {H}$ attached to an Eichler order of squarefree level in an indefinite quaternion algebra over $\mathbb {Q}$ . Denote by V the hyperbolic volume of said surface. We prove the sup-norm estimate $$\begin{align*}\| \Im(\cdot)^{\frac{k}{2}} f \|_{\infty} \ll_{\varepsilon} (k V)^{\frac{1}{4}+\varepsilon} \end{align*}$$ with absolute implied constant. For a cuspidal Maaß newform $\varphi $ of eigenvalue $\lambda $ on such a surface, we prove that $$\begin{align*}\|\varphi \|_{\infty} \ll_{\lambda,\varepsilon} V^{\frac{1}{4}+\varepsilon}. \end{align*}$$ We establish analogous estimates in the setting of definite quaternion algebras.

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Section: Introductionmentioning

confidence: 94%

Theta functions, fourth moments of eigenforms and the sup-norm problem II

Khayutin,

Nelson,

Steiner

2024

Forum of Mathematics, Pi

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“…For squarefree level, the estimate (1.5) is comparable in strength to the Weyl bound for the Riemann zeta function, and has long been regarded as a natural limit for the sup-norm problem in the squarefree level aspect [HT13,Remarks (i)]. It has been extended in various ways -to number fields [BHM16, BHMM20,Ass17], to levels N that are not necessarily squarefree [Sah17, Sah20, Com21, HS20], and to more general vectors than newforms [HNS17,Ass21].…”

Section: Introductionmentioning

confidence: 99%

Theta functions, fourth moments of eigenforms, and the sup-norm problem II

Khayutin¹,

Nelson²,

Steiner³

2022

Preprint

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For an L 2 -normalized holomorphic newform f of weight k on a hyperbolic surface of volume V attached to an Eichler order of squarefree level in an indefinite quaternion algebra over Q, we prove the sup-norm estimate Im(•)with absolute implied constant. For a cuspidal Maaß newform ϕ of eigenvalue λ on such a surface, we prove thatWe establish analogous estimates in the setting of definite quaternion algebras. Contents1. Introduction 1 2. Statement of results 8 3. Division and reduction of the proof 11 4. Arithmetic quotients as real manifolds 16 5. Theta kernels and their L 2 -norms 18 6. Preliminaries on the geometry of numbers 25 7. Local preliminaries on orders 27 8. Invariants of rational quadratic forms 29 9. Type I estimates 31 10. Type II estimates 33 Appendix A. The Theta Lift 37 References 48

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The sup-norm problem beyond the newform

Cited by 2 publications

References 12 publications

Theta functions, fourth moments of eigenforms and the sup-norm problem II

Theta functions, fourth moments of eigenforms and the sup-norm problem II

Theta functions, fourth moments of eigenforms, and the sup-norm problem II

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