$L^p$-Norm Bounds for Automorphic Forms via Spectral Reciprocity
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| Natura: | Preprint |
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2022
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| _version_ | 1866913910522642432 |
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| author | Humphries, Peter Khan, Rizwanur |
| author_facet | Humphries, Peter Khan, Rizwanur |
| contents | Let $g$ be a Hecke-Maass cusp form on the modular surface ${\rm SL}_2(\mathbb{Z})\backslash\mathbb{H}$, namely an $L^2$-normalised nonconstant Laplacian eigenfunction on ${\rm SL}_2(\mathbb{Z})\backslash\mathbb{H}$ that is additionally a joint eigenfunction of every Hecke operator. We prove the $L^4$-norm bound $\|g\|_4\ll_{\varepsilon}λ_g^{3/304+\varepsilon}$, where $λ_g$ denotes the Laplacian eigenvalue of $g$, which improves upon Sogge's $L^4$-norm bound $\|g\|_4\llλ_g^{1/16}$ for Laplacian eigenfunctions on a compact Riemann surface by more than a six-fold power-saving. Via interpolation, this yields $L^p$-norm bounds for Hecke-Maass cusp forms that are power-saving improvements on Sogge's bounds for all $p>2$. Our paper marks the first improvement of Sogge's result on the modular surface. Furthermore, these methods yield for compact arithmetic surfaces the best $L^4$-norm bound to date.
Via the Watson-Ichino triple product formula, bounds for the $L^4$-norm of $g$ are reduced to bounds for certain mixed moments of $L$-functions. We bound these using two forms of spectral reciprocity. The first is a form of ${\rm GL}_3\times{\rm GL}_2\leftrightsquigarrow{\rm GL}_4\times{\rm GL}_1$ spectral reciprocity, which relates a ${\rm GL}_2$ moment of ${\rm GL}_3\times{\rm GL}_2$ Rankin-Selberg $L$-functions to a ${\rm GL}_1$ moment of ${\rm GL}_4\times{\rm GL}_1$ Rankin-Selberg $L$-functions; this can be seen as a cuspidal analogue of Motohashi's formula relating the fourth moment of the Riemann zeta function to the third moment of central values of Hecke $L$-functions. The second is a form of ${\rm GL}_4\times{\rm GL}_2\leftrightsquigarrow{\rm GL}_4\times{\rm GL}_2$ spectral reciprocity, which is a cuspidal analogue of a formula of Kuznetsov for the fourth moment of central values of Hecke $L$-functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_05613 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | $L^p$-Norm Bounds for Automorphic Forms via Spectral Reciprocity Humphries, Peter Khan, Rizwanur Number Theory Analysis of PDEs 11F12 (primary), 11F66, 11M41, 58J51, 81Q50 (secondary) Let $g$ be a Hecke-Maass cusp form on the modular surface ${\rm SL}_2(\mathbb{Z})\backslash\mathbb{H}$, namely an $L^2$-normalised nonconstant Laplacian eigenfunction on ${\rm SL}_2(\mathbb{Z})\backslash\mathbb{H}$ that is additionally a joint eigenfunction of every Hecke operator. We prove the $L^4$-norm bound $\|g\|_4\ll_{\varepsilon}λ_g^{3/304+\varepsilon}$, where $λ_g$ denotes the Laplacian eigenvalue of $g$, which improves upon Sogge's $L^4$-norm bound $\|g\|_4\llλ_g^{1/16}$ for Laplacian eigenfunctions on a compact Riemann surface by more than a six-fold power-saving. Via interpolation, this yields $L^p$-norm bounds for Hecke-Maass cusp forms that are power-saving improvements on Sogge's bounds for all $p>2$. Our paper marks the first improvement of Sogge's result on the modular surface. Furthermore, these methods yield for compact arithmetic surfaces the best $L^4$-norm bound to date. Via the Watson-Ichino triple product formula, bounds for the $L^4$-norm of $g$ are reduced to bounds for certain mixed moments of $L$-functions. We bound these using two forms of spectral reciprocity. The first is a form of ${\rm GL}_3\times{\rm GL}_2\leftrightsquigarrow{\rm GL}_4\times{\rm GL}_1$ spectral reciprocity, which relates a ${\rm GL}_2$ moment of ${\rm GL}_3\times{\rm GL}_2$ Rankin-Selberg $L$-functions to a ${\rm GL}_1$ moment of ${\rm GL}_4\times{\rm GL}_1$ Rankin-Selberg $L$-functions; this can be seen as a cuspidal analogue of Motohashi's formula relating the fourth moment of the Riemann zeta function to the third moment of central values of Hecke $L$-functions. The second is a form of ${\rm GL}_4\times{\rm GL}_2\leftrightsquigarrow{\rm GL}_4\times{\rm GL}_2$ spectral reciprocity, which is a cuspidal analogue of a formula of Kuznetsov for the fourth moment of central values of Hecke $L$-functions. |
| title | $L^p$-Norm Bounds for Automorphic Forms via Spectral Reciprocity |
| topic | Number Theory Analysis of PDEs 11F12 (primary), 11F66, 11M41, 58J51, 81Q50 (secondary) |
| url | https://arxiv.org/abs/2208.05613 |