The Twelfth Moment of Hecke $L$-Functions in the Weight Aspect
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866916309913042944 |
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| author | Humphries, Peter Khan, Rizwanur |
| author_facet | Humphries, Peter Khan, Rizwanur |
| contents | We prove an upper bound for the twelfth moment of Hecke $L$-functions associated to holomorphic Hecke cusp forms of weight $k$ in a dyadic interval $T \leq k \leq 2T$ as $T$ tends to infinity. This bound recovers the Weyl-strength subconvex bound $L(1/2,f) \ll_{\varepsilon} k^{1/3 + \varepsilon}$ and shows that for any $δ> 0$, the sub-Weyl subconvex bound $L(1/2,f) \ll k^{1/3 - δ}$ holds for all but $O_{\varepsilon}(T^{12δ+ \varepsilon})$ Hecke cusp forms $f$ of weight at most $T$. Our result parallels a related result of Jutila for the twelfth moment of Hecke $L$-functions associated to Hecke-Maass cusp forms. The proof uses in a crucial way a spectral reciprocity formula of Kuznetsov that relates the fourth moment of $L(1/2,f)$ weighted by a test function to a dual fourth moment weighted by a different test function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_00543 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The Twelfth Moment of Hecke $L$-Functions in the Weight Aspect Humphries, Peter Khan, Rizwanur Number Theory 11F66 (primary), 11F11 (secondary) We prove an upper bound for the twelfth moment of Hecke $L$-functions associated to holomorphic Hecke cusp forms of weight $k$ in a dyadic interval $T \leq k \leq 2T$ as $T$ tends to infinity. This bound recovers the Weyl-strength subconvex bound $L(1/2,f) \ll_{\varepsilon} k^{1/3 + \varepsilon}$ and shows that for any $δ> 0$, the sub-Weyl subconvex bound $L(1/2,f) \ll k^{1/3 - δ}$ holds for all but $O_{\varepsilon}(T^{12δ+ \varepsilon})$ Hecke cusp forms $f$ of weight at most $T$. Our result parallels a related result of Jutila for the twelfth moment of Hecke $L$-functions associated to Hecke-Maass cusp forms. The proof uses in a crucial way a spectral reciprocity formula of Kuznetsov that relates the fourth moment of $L(1/2,f)$ weighted by a test function to a dual fourth moment weighted by a different test function. |
| title | The Twelfth Moment of Hecke $L$-Functions in the Weight Aspect |
| topic | Number Theory 11F66 (primary), 11F11 (secondary) |
| url | https://arxiv.org/abs/2207.00543 |