Product of Rankin-Selberg convolutions and a new proof of Jacquet's local converse conjecture
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866917747984695296 |
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| author | Yan, Pan Zhang, Qing |
| author_facet | Yan, Pan Zhang, Qing |
| contents | In this article, we construct a family of integrals which represent the product of Rankin-Selberg $L$-functions of $\mathrm{GL}_{l}\times \mathrm{GL}_m$ and of $\mathrm{GL}_{l}\times \mathrm{GL}_n $ when $m+n<l$. When $n=0$, these integrals are those defined by Jacquet--Piatetski-Shapiro--Shalika up to a shift. In this sense, these new integrals generalize Jacquet--Piatetski-Shapiro--Shalika's Rankin-Selberg convolution integrals. We study basic properties of these integrals. In particular, we define local gamma factors using this new family of integrals. As an application, we obtain a new proof of Jacquet's local converse conjecture using these new integrals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_10445 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Product of Rankin-Selberg convolutions and a new proof of Jacquet's local converse conjecture Yan, Pan Zhang, Qing Representation Theory Number Theory 11F70, 22E50 In this article, we construct a family of integrals which represent the product of Rankin-Selberg $L$-functions of $\mathrm{GL}_{l}\times \mathrm{GL}_m$ and of $\mathrm{GL}_{l}\times \mathrm{GL}_n $ when $m+n<l$. When $n=0$, these integrals are those defined by Jacquet--Piatetski-Shapiro--Shalika up to a shift. In this sense, these new integrals generalize Jacquet--Piatetski-Shapiro--Shalika's Rankin-Selberg convolution integrals. We study basic properties of these integrals. In particular, we define local gamma factors using this new family of integrals. As an application, we obtain a new proof of Jacquet's local converse conjecture using these new integrals. |
| title | Product of Rankin-Selberg convolutions and a new proof of Jacquet's local converse conjecture |
| topic | Representation Theory Number Theory 11F70, 22E50 |
| url | https://arxiv.org/abs/2309.10445 |