A geometric approach to the sup‐norm problem for automorphic forms: the case of newforms on GL2(Fq(T)) with squarefree level

Sawin, Will

doi:10.1112/plms.12389

Cited by 3 publications

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“…Remark 1.5. In a function field setting analogous to that of Theorem 1.1, Sawin [Saw21] has used geometric techniques to establish (among other things) the sup-norm bound 𝑁 1 4 +𝛼 𝑞 , where 𝛼 𝑞 > 0 tends to zero as the cardinality q of the underlying finite field tends to ∞. We do not see any obstruction to adapting the techniques of this paper to the function field setting, where we expect they would give the improved bound 𝜀 𝑁 1 4 +𝜀 .…”

Section: Introductionmentioning

confidence: 99%

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: 99%

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

Khayutin,

Nelson,

Steiner

2024

Forum of Mathematics, Pi

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“…Remark 1.5. In a function field setting analogous to that of Theorem 1.1, Sawin [Saw21] has used geometric techniques to establish (among other things) the sup-norm bound ≪ N 1 4 +αq , where α q > 0 tends to zero as the cardinality q of the underlying finite field tends to ∞. We do not see any obstruction to adapting the techniques of this paper to the function field setting, where we expect they would give the improved bound ≪ ε N 1 4 +ε .…”

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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Sup-norms of eigenfunctions in the level aspect for compact arithmetic surfaces, II: newforms and subconvexity

Saha

2019

Preprint

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We obtain for the first time an improvement over the local bound in the depth aspect for sup-norms of newforms on an indefinite quaternion division algebra over Q. A central role in our method is played by the decay of local matrix coefficients. More generally, we prove a strong upper bound in the level aspect for sup-norms of automorphic forms belonging to any family whose associated matrix coefficients have a proper decay along a suitable sequence of compact subsets.Remark 1.1. The lists above omit numerous recent works on the sup-norm problem concerning lower bounds, non-trivial character, hybrid bounds, holomorphic modular forms, real weights, generalizations to number fields and to higher rank groups, forms that do not correspond to newvectors at the ramified places, and so on and so forth. We refer the reader to the introductions of [2,17] for brief discussions of some of these related results. The bound(1)2+ǫ 1 is of particular importance because it is the immediate bound emerging from the adelic pre-trace formula where the local test function at each ramified prime is chosen to be essentially the matrix coefficient of a (suitable translate of a) local newvector. This bound was originally proved by Marshall [14] for D a division algebra; in the case where D = M 2 (Q), it was noted in [14] that the same bound holds provided one restricts the domain to a fixed compact set. With additional work involving Whittaker expansions near various cusps, which was done in [16], it is now known that the bound (1) also holds for D = M 2 (Q). In fact, as noted earlier, it was shown in [16] that the stronger bound f ∞ ≪ λ,ǫ N 1/6 N

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A geometric approach to the sup‐norm problem for automorphic forms: the case of newforms on GL2(Fq(T)) with squarefree level

Cited by 3 publications

References 21 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

Sup-norms of eigenfunctions in the level aspect for compact arithmetic surfaces, II: newforms and subconvexity

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