Large values of cusp forms on $$\mathrm {GL}_n$$

Brumley, Farrell; Templier, Nicolas

doi:10.1007/s00029-020-00589-z

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Sup-norms Of Automorphic Forms1

Introduction1

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2022

2023

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Cited by 4 publications

(2 citation statements)

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“…there are known phenomena where (1.7) is violated. One is the behaviour close to the cusps [6,24] if the underlying space is not compact. This is an artefact of special functions that we do not focus on in this discussion.…”

Section: Sup-norms Of Automorphic Formsmentioning

confidence: 99%

Epstein Zeta-Functions, Subconvexity, and the Purity Conjecture

Blomer¹

2018

J. Inst. Math. Jussieu

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Section: Sup-norms Of Automorphic Formsmentioning

confidence: 99%

Epstein Zeta-Functions, Subconvexity, and the Purity Conjecture

Blomer¹

2018

J. Inst. Math. Jussieu

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“…Proposition 4.5) suggest that this is attained 'high in the cusp', at least in the average. This can be compared with the work of Brumley-Templier [BT20]. From Theorem 1.3, we see that the supremum of B k (Z, Z) grows as at least k 3n(n+1)/4 , Theorem 4.5 and the second bound in Theorem 7.5 show that such a value can only be attained in the following region:…”

Section: Introductionmentioning

confidence: 80%

Bounds for the Bergman kernel and the sup-norm of holomorphic Siegel cusp forms

Das¹,

Krishna²

2022

Preprint

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We prove 'polynomial in k' bounds on the size of the Bergman kernel for the space of holomorphic Siegel cusp forms of degree n and weight k. When n = 1, 2 our bounds agree with the conjectural bounds on the aforementioned size, while the lower bounds match for all n ≥ 1. For an L 2 -normalised Siegel cusp form F of degree 2, our bound for its supnorm is O ǫ (k 9/4+ǫ ). Further, we show that in any compact set Ω (which does not depend on k) contained in the Siegel fundamental domain of Sp(2, Z) on the Siegel upper half space, the sup-norm of F is O Ω (k 3/2−η ) for some η > 0, going beyond the 'generic' bound in this setting.

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The sup-norm problem for automorphic cusp forms of 𝑃𝐺𝐿(𝑛,ℤ[𝕚])

Maga¹,

Zábrádi²

2023

Proc. Amer. Math. Soc.

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Let ϕ \phi be an L 2 L^2 -normalized Hecke–Maaß cusp form on the locally symmetric space X ≔ P G L n ( Z [ i ] ) ∖ P G L n ( C ) / P U n X≔\mathrm {PGL}_n(\mathbb {Z}[i])\backslash \mathrm {PGL}_n(\mathbb {C}) / \mathrm {PU}_n . If Ω \Omega is a compact subset of X X , then we prove the bound ‖ ϕ | Ω ‖ ∞ ≪ Ω λ ϕ n ( n − 1 ) / 4 − δ \|\phi |_{\Omega }\|_{\infty }\ll _{\Omega } \lambda _{\phi }^{n(n-1)/4-\delta } for some δ > 0 \delta >0 depending only on n n , where λ ϕ \lambda _{\phi } is the Laplace eigenvalue of ϕ \phi .

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Large values of cusp forms on $$\mathrm {GL}_n$$

Cited by 4 publications

References 62 publications

Epstein Zeta-Functions, Subconvexity, and the Purity Conjecture

Epstein Zeta-Functions, Subconvexity, and the Purity Conjecture

Bounds for the Bergman kernel and the sup-norm of holomorphic Siegel cusp forms

The sup-norm problem for automorphic cusp forms of 𝑃𝐺𝐿(𝑛,ℤ[𝕚])

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