The orbit method in number theory through the sup-norm problem for $\operatorname{GL}(2)$
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| Format: | Preprint |
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2025
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| _version_ | 1866908719022866432 |
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| author | Assing, Edgar Toma, Radu |
| author_facet | Assing, Edgar Toma, Radu |
| contents | 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 $φ$ of large prime-power level $N = p^{4n}$ and large eigenvalue $λ$. It states that $\| φ\|_\infty \ll_p (λN)^{5/24 + \varepsilon}$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_06224 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The orbit method in number theory through the sup-norm problem for $\operatorname{GL}(2)$ Assing, Edgar Toma, Radu Number Theory 11F70 (Primary), 11F12, 11D45, 35A27, 53D50 (Secondary) 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 $φ$ of large prime-power level $N = p^{4n}$ and large eigenvalue $λ$. It states that $\| φ\|_\infty \ll_p (λN)^{5/24 + \varepsilon}$, 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. |
| title | The orbit method in number theory through the sup-norm problem for $\operatorname{GL}(2)$ |
| topic | Number Theory 11F70 (Primary), 11F12, 11D45, 35A27, 53D50 (Secondary) |
| url | https://arxiv.org/abs/2503.06224 |