Critical values of $L$-functions of residual representations of $\mathrm{GL}_4$
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866917392088563712 |
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| author | Droschl, Johannes |
| author_facet | Droschl, Johannes |
| contents | In this paper we prove rationality results of critical values for $L$-functions attached to representations in the residual spectrum of $\mathrm{GL}_4(\mathbb{A})$. We use the Jacquet-Langlands correspondence to describe their partial $L$-functions via cuspidal automorphic representations of the group $\mathrm{GL}_2'(\mathbb{A})$ over a quaternion algebra. Using ideas inspired by results of Grobner and Raghuram we are then able to compute the critical values as a Shalika period up to a rational multiple. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_13464 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Critical values of $L$-functions of residual representations of $\mathrm{GL}_4$ Droschl, Johannes Number Theory 11F67, 11F70, 11F75 In this paper we prove rationality results of critical values for $L$-functions attached to representations in the residual spectrum of $\mathrm{GL}_4(\mathbb{A})$. We use the Jacquet-Langlands correspondence to describe their partial $L$-functions via cuspidal automorphic representations of the group $\mathrm{GL}_2'(\mathbb{A})$ over a quaternion algebra. Using ideas inspired by results of Grobner and Raghuram we are then able to compute the critical values as a Shalika period up to a rational multiple. |
| title | Critical values of $L$-functions of residual representations of $\mathrm{GL}_4$ |
| topic | Number Theory 11F67, 11F70, 11F75 |
| url | https://arxiv.org/abs/2407.13464 |