Spectral Reciprocity: A Fourier--Analytic Approach
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909941666676736 |
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| author | Yang, Liyang |
| author_facet | Yang, Liyang |
| contents | We develop a Fourier--analytic framework for establishing spectral reciprocity formulas linking $\mathrm{GL}_3$ and $\mathrm{GL}_2$ automorphic spectra over number fields. The method applies uniformly to cuspidal and non-cuspidal $\mathrm{GL}_3$ representations and treats Motohashi-type and Blomer--Khan-type reciprocities in a parallel manner, revealing intrinsic connections between them and extending each to new settings. We also obtain explicit weight transforms in the analytic newvector and spherical cases. Applications include first-moment estimates for $\mathrm{GL}_3\times\mathrm{GL}_2$ $L$-functions over number fields, an explicit twisted fourth moment for $\mathrm{GL}_2$ $L$-functions over totally real fields, a sharp upper bound for the fifth moment, subconvexity for triple product $L$-functions, and new simultaneous nonvanishing results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_03305 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral Reciprocity: A Fourier--Analytic Approach Yang, Liyang Number Theory We develop a Fourier--analytic framework for establishing spectral reciprocity formulas linking $\mathrm{GL}_3$ and $\mathrm{GL}_2$ automorphic spectra over number fields. The method applies uniformly to cuspidal and non-cuspidal $\mathrm{GL}_3$ representations and treats Motohashi-type and Blomer--Khan-type reciprocities in a parallel manner, revealing intrinsic connections between them and extending each to new settings. We also obtain explicit weight transforms in the analytic newvector and spherical cases. Applications include first-moment estimates for $\mathrm{GL}_3\times\mathrm{GL}_2$ $L$-functions over number fields, an explicit twisted fourth moment for $\mathrm{GL}_2$ $L$-functions over totally real fields, a sharp upper bound for the fifth moment, subconvexity for triple product $L$-functions, and new simultaneous nonvanishing results. |
| title | Spectral Reciprocity: A Fourier--Analytic Approach |
| topic | Number Theory |
| url | https://arxiv.org/abs/2512.03305 |