Bessel Periods on $U(2,1) \times U(1,1)$, Relative Trace Formula and Non-Vanishing of Central $L$-values
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| Format: | Preprint |
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2023
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| _version_ | 1866908438387228672 |
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| author | Michel, Philippe Ramakrishnan, Dinakar Yang, Liyang |
| author_facet | Michel, Philippe Ramakrishnan, Dinakar Yang, Liyang |
| contents | In this paper we calculate the asymptotics of the second moment of the Bessel periods associated to certain holomorphic cuspidal representations $(π, π')$ of $U(2,1) \times U(1,1)$ of regular infinity type (averaged over $π$). Using these, we obtain quantitative non-vanishing results for the Rankin-Selberg central $L$-values $L(1/2, π\times π')$, which are of degree twelve over $\mathbb{Q}$, with concomitant difficulty in applying standard methods, especially since we are in a `conductor dropping' situation. We use the relative trace formula, and the orbital integrals are evaluated rather than compared with others. Besides their intrinsic interest, non-vanishing of these critical values also lead, by known results, to deducing certain associated Selmer groups have rank zero. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_08490 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Bessel Periods on $U(2,1) \times U(1,1)$, Relative Trace Formula and Non-Vanishing of Central $L$-values Michel, Philippe Ramakrishnan, Dinakar Yang, Liyang Number Theory 11F67, 11F70, 11F72 In this paper we calculate the asymptotics of the second moment of the Bessel periods associated to certain holomorphic cuspidal representations $(π, π')$ of $U(2,1) \times U(1,1)$ of regular infinity type (averaged over $π$). Using these, we obtain quantitative non-vanishing results for the Rankin-Selberg central $L$-values $L(1/2, π\times π')$, which are of degree twelve over $\mathbb{Q}$, with concomitant difficulty in applying standard methods, especially since we are in a `conductor dropping' situation. We use the relative trace formula, and the orbital integrals are evaluated rather than compared with others. Besides their intrinsic interest, non-vanishing of these critical values also lead, by known results, to deducing certain associated Selmer groups have rank zero. |
| title | Bessel Periods on $U(2,1) \times U(1,1)$, Relative Trace Formula and Non-Vanishing of Central $L$-values |
| topic | Number Theory 11F67, 11F70, 11F72 |
| url | https://arxiv.org/abs/2309.08490 |