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arXiv · 2401.02741

The sup-norm problem for newforms of large level on $\operatorname{PGL}(n)$

Radu Toma
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Abstract

Let $N$ be a prime and $\phi$ be a Hecke-Maass cuspidal newform for the Hecke congruence subgroup $\Gamma_0(N)$ in $\operatorname{SL}_n(\mathbb{R})$. Let $\Omega$ be an adelic compactum and let $\Omega_N$ be its projection to $\Gamma_0(N) \backslash \operatorname{SL}_n(\mathbb{R}) / \operatorname{SO}(n)$. For any prime $n$, we prove sub-baseline bounds for the sup-norm of $\phi$ restricted to $\Omega_N$. Conditionally on GRH, we generalise this result to all $n \geq 2$. The methods involve a new reduction theory with level structure, based on generalisations of Atkin-Lehner operators.

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Radu Toma. 2024-01-05. The sup-norm problem for newforms of large level on $\operatorname{PGL}(n)$. https://arxiv.org/abs/2401.02741

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