Subpolynomial compact-set peaks of Hecke–Maass waves
Let $X = \operatorname{PSL}_2(\mathbb{Z}) \backslash \mathbb{H}$ carry the hyperbolic measure $dx\,dy/y^2$. Let $u$ be an $L^2$-normalised smooth real cusp form with $$-y^2(\partial_x^2 + \partial_y^2)\,u = \bigl(\tfrac14 + t^2\bigr)\,u, \qquad t \ge 0,$$ which is an eigenfunction of every Hecke operator $$(T_n f)(z) = n^{-1/2} \sum_{ad = n,\ a, d > 0}\ \sum_{b=0}^{d-1} f\!\left(\frac{az + b}{d}\right);$$ the cusp condition means $\int_0^1 u(x + iy)\,dx = 0$ for every $y > 0$. The general bound is $\sup_X |u| = O\bigl((1+t)^{1/2}\bigr)$, and Iwaniec and Sarnak improved it to $\sup_X |u| = O_\varepsilon\bigl((1+t)^{5/12+\varepsilon}\bigr)$. Over all of $X$ no bound of the form below holds: $\sup_X |u| \ge c_\varepsilon (1+t)^{1/6-\varepsilon}$, from the behaviour of $u$ high in the cusp (Sarnak; see Petridis and Risager).
Problem. Is it true that for each compact $K \subset X$ and every $\varepsilon > 0$ there is a constant $C_{K,\varepsilon}$, independent of $u$ and $t$, such that $$\sup_K |u| \le C_{K,\varepsilon}\,(1+t)^{\varepsilon}$$ for every such $u$? The compact set is fixed as $t$ grows.
Source: Iwaniec and Sarnak (1995); stated for compact sets in Sarnak (2003), Conjecture 4, for eigenfunctions on hyperbolic surfaces; in this form, for Hecke–Maass cusp forms on the modular surface, Petridis and Risager (2018), conjecture (C2).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- NT — Number Theory
- Secondary
- SP
- Keywords
- AIM Open Applied Problems, arithmetic quantum chaos, Maass forms, sup norms
References
@article{iwaniec1995,
author = {Iwaniec, Henryk and Sarnak, Peter},
title = {{$L^\infty$} norms of eigenfunctions of arithmetic surfaces},
journal = {Ann. of Math. (2)},
volume = {141},
number = {2},
year = {1995},
pages = {301--320},
doi = {10.2307/2118522},
}
@article{sarnak2003,
author = {Sarnak, Peter},
title = {Spectra of hyperbolic surfaces},
journal = {Bull. Amer. Math. Soc. (N.S.)},
volume = {40},
number = {4},
year = {2003},
pages = {441--478},
doi = {10.1090/S0273-0979-03-00991-1},
}
@article{petridis2018,
author = {Petridis, Yiannis N. and Risager, Morten S.},
title = {Averaging over {H}eegner points in the hyperbolic circle problem},
journal = {Int. Math. Res. Not. IMRN},
volume = {2018},
number = {16},
year = {2018},
pages = {4942--4968},
doi = {10.1093/imrn/rnx026},
eprint = {1610.09393},
archivePrefix = {arXiv},
}
@misc{aim-sarnak-compact-supnorm,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Subpolynomial compact-set peaks of Hecke–Maass waves}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/118-sarnak-compact-supnorm.md},
note = {Entry 118 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00379, "Subpolynomial compact-set peaks of Hecke–Maass waves", clopen.solutions.