Period relations between the Betti-Whittaker periods for ${\rm GL}_n$ under duality
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866916263045890048 |
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| author | Chen, Shih-Yu |
| author_facet | Chen, Shih-Yu |
| contents | In this paper, under some regularity conditions, we prove a period relation between the Betti--Whittaker periods associated to a regular algebraic cuspidal automorphic representation of ${\rm GL}_n(\mathbb{A})$ and its contragredient. As a consequence, we obtain the trivialness of the relative period associated to a regular algebraic cuspidal automorphic representation of ${\rm GL}_{2n}(\mathbb{A})$ of orthogonal type, which implies the algebraicity of the ratios of successive critical $L$-values for ${\rm GSpin}_{2n}^* \times {\rm GL}_{n'}$ by the result of Harder and Raghuram. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_04714 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Period relations between the Betti-Whittaker periods for ${\rm GL}_n$ under duality Chen, Shih-Yu Number Theory In this paper, under some regularity conditions, we prove a period relation between the Betti--Whittaker periods associated to a regular algebraic cuspidal automorphic representation of ${\rm GL}_n(\mathbb{A})$ and its contragredient. As a consequence, we obtain the trivialness of the relative period associated to a regular algebraic cuspidal automorphic representation of ${\rm GL}_{2n}(\mathbb{A})$ of orthogonal type, which implies the algebraicity of the ratios of successive critical $L$-values for ${\rm GSpin}_{2n}^* \times {\rm GL}_{n'}$ by the result of Harder and Raghuram. |
| title | Period relations between the Betti-Whittaker periods for ${\rm GL}_n$ under duality |
| topic | Number Theory |
| url | https://arxiv.org/abs/2302.04714 |