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arXiv · 2503.06224

The orbit method in number theory through the sup-norm problem for $\operatorname{GL}(2)$

Edgar Assing·Radu Toma
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Abstract

The orbit method in its quantitative form due to Nelson and Venkatesh has played a central role in recent advances in the analytic theory of higher rank $L$-functions. The goal of this note is to explain how the method can be applied to the sup-norm problem for automorphic forms on $\operatorname{PGL}(2)$. Doing so, we prove a new hybrid bound for newforms $\varphi$ of large prime-power level $N = p^{4n}$ and large eigenvalue $\lambda$. It states that $\| \varphi \|_\infty \ll_p (\lambda N)^{5/24 + \varepsilon}$, recovering the result of Iwaniec and Sarnak spectrally and improving the local bound in the depth aspect for the first time in this non-compact setting. We also provide an exposition of the microlocal tools used, illustrating and motivating the theory through the classical case of $\operatorname{PGL}(2)$, following notes and lectures of Nelson and Venkatesh.

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Edgar Assing, Radu Toma. 2025-03-08. The orbit method in number theory through the sup-norm problem for $\operatorname{GL}(2)$. https://arxiv.org/abs/2503.06224

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