Finite period vectors and Gauss sums
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866914742722887680 |
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| author | Jo, Yeongseong |
| author_facet | Jo, Yeongseong |
| contents | We study four sums including the Jacquet--Piatetski-Shapiro--Shalika, Flicker, Bump--Friedberg, and Jacquet--Shalika sums associated to irreducible cuspidal representations of general linear groups over finite fields. By computing explicitly, we relate Asai and Bump--Friedberg gamma factors over finite fields to those over nonarchimedean local fields through level zero supercuspidal representation. Via Deligne--Kazhdan close field theory, we prove that exterior square and Bump--Friedberg gamma factors agree with corresponding Artin gamma factors of their associated tamely ramified representations through local Langlands correspondence. We also deduce product formulae for Asai, Bump--Friedberg, and exterior square gamma factors in terms of Gauss sums. By combining these results, we examine Jacquet--Piatetski-Shapiro--Shalika, Flicker--Rallis, Jacquet--Shalika, and Friedberg--Jacquet periods and vectors and their connections to Rankin-Selberg, Asai, exterior square, and Bump-Friedberg gamma factors, respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_02085 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Finite period vectors and Gauss sums Jo, Yeongseong Number Theory Representation Theory We study four sums including the Jacquet--Piatetski-Shapiro--Shalika, Flicker, Bump--Friedberg, and Jacquet--Shalika sums associated to irreducible cuspidal representations of general linear groups over finite fields. By computing explicitly, we relate Asai and Bump--Friedberg gamma factors over finite fields to those over nonarchimedean local fields through level zero supercuspidal representation. Via Deligne--Kazhdan close field theory, we prove that exterior square and Bump--Friedberg gamma factors agree with corresponding Artin gamma factors of their associated tamely ramified representations through local Langlands correspondence. We also deduce product formulae for Asai, Bump--Friedberg, and exterior square gamma factors in terms of Gauss sums. By combining these results, we examine Jacquet--Piatetski-Shapiro--Shalika, Flicker--Rallis, Jacquet--Shalika, and Friedberg--Jacquet periods and vectors and their connections to Rankin-Selberg, Asai, exterior square, and Bump-Friedberg gamma factors, respectively. |
| title | Finite period vectors and Gauss sums |
| topic | Number Theory Representation Theory |
| url | https://arxiv.org/abs/2307.02085 |