Simultaneous non-vanishing of central values of $\mathrm{GL}(2)\times \mathrm{GL}(3)$ and $\mathrm{GL}(3)\times \mathrm{GL}(3)$ $L$-functions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911561472278528 |
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| author | Pan, Junjie |
| author_facet | Pan, Junjie |
| contents | Let $g$ denote a fixed holomorphic Hecke cusp form of weight $k \equiv 0 \pmod{4}$ on $\mathrm{SL}_2(\mathbb{Z})$, and let $π$ be a fixed cuspidal automorphic representation of $\mathrm{GL}_3$. In this paper, we establish an asymptotic formula for the first moment of the product \[L(1/2,g\times F)L(1/2,π\times F),\] where $F$ runs over an orthonormal basis of Hecke-Maaß cusp forms of level $q$ on $\mathrm{GL}_3$. As an application, we deduce that $L(1/2, g\times F)L(1/2,π\times F)\neq 0$ for infinitely many such forms $F$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_15704 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Simultaneous non-vanishing of central values of $\mathrm{GL}(2)\times \mathrm{GL}(3)$ and $\mathrm{GL}(3)\times \mathrm{GL}(3)$ $L$-functions Pan, Junjie Number Theory 11F70(Automorphic forms, L-functions), 11M41(L-functions, special values) Let $g$ denote a fixed holomorphic Hecke cusp form of weight $k \equiv 0 \pmod{4}$ on $\mathrm{SL}_2(\mathbb{Z})$, and let $π$ be a fixed cuspidal automorphic representation of $\mathrm{GL}_3$. In this paper, we establish an asymptotic formula for the first moment of the product \[L(1/2,g\times F)L(1/2,π\times F),\] where $F$ runs over an orthonormal basis of Hecke-Maaß cusp forms of level $q$ on $\mathrm{GL}_3$. As an application, we deduce that $L(1/2, g\times F)L(1/2,π\times F)\neq 0$ for infinitely many such forms $F$. |
| title | Simultaneous non-vanishing of central values of $\mathrm{GL}(2)\times \mathrm{GL}(3)$ and $\mathrm{GL}(3)\times \mathrm{GL}(3)$ $L$-functions |
| topic | Number Theory 11F70(Automorphic forms, L-functions), 11M41(L-functions, special values) |
| url | https://arxiv.org/abs/2510.15704 |