Critical values of $L$-functions of residual representations of $\mathrm{GL}_4$

[2407.13464] Critical values of $L$-functions of residual representations of $\mathrm{GL}_4$ Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Number Theory arXiv:2407.13464 (math) [Submitted on 18 Jul 2024 ( v1 ), last revised 8 Apr 2026 (this version, v3)] Title: Critical values of $L$-functions of residual representations of $\mathrm{GL}_4$ Authors: Johannes Droschl View a PDF of the paper titled Critical values of $L$-functions of residual representations of $\mathrm{GL}_4$, by Johannes Droschl View PDF HTML (experimental) Abstract: 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. Comments: Added clarifications and fixed a minor mistake in Section 7 Subjects: Number Theory (math.NT) MSC classes: 11F67, 11F70, 11F75 Cite as: arXiv:2407.13464 [math.NT] (or arXiv:2407.13464v3 [math.NT] for this version) https://doi.org/10.48550/arXiv.2407.13464 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Johannes Droschl [ view email ] [v1] Thu, 18 Jul 2024 12:37:35 UTC (54 KB) [v2] Thu, 20 Feb 2025 07:27:28 UTC (66 KB) [v3] Wed, 8 Apr 2026 19:11:04 UTC (55 KB) Full-text links: Access Paper: View a PDF of the paper titled Critical values of $L$-functions of residual representations of $\mathrm{GL}_4$, by Johannes Droschl View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2024-07 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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