Beyond endoscopy for $\mathsf{GL}_2$ over $\mathbb{Q}$ with ramification 2: bounds towards the Ramanujan conjecture

[2507.09655] Beyond endoscopy for $\mathsf{GL}_2$ over $\mathbb{Q}$ with ramification 2: bounds towards the Ramanujan conjecture Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Number Theory arXiv:2507.09655 (math) [Submitted on 13 Jul 2025 ( v1 ), last revised 15 Aug 2026 (this version, v3)] Title: Beyond endoscopy for $\mathsf{GL}_2$ over $\mathbb{Q}$ with ramification 2: bounds towards the Ramanujan conjecture Authors: Yuhao Cheng View a PDF of the paper titled Beyond endoscopy for $\mathsf{GL}_2$ over $\mathbb{Q}$ with ramification 2: bounds towards the Ramanujan conjecture, by Yuhao Cheng View PDF HTML (experimental) Abstract: We continue generalizing Altuğ's work on $\mathsf{GL}_2$ over $\mathbb{Q}$ in the unramified setting for \emph{Beyond Endoscopy} to the ramified case where ramification occurs at $S=\{\infty,q_1,\dots,q_r\}$ with $2\in S$, after generalizing the first step. We establish a new proof of the $1/4$ bound towards the Ramanujan conjecture for the trace of the cuspidal part in the ramified case, which is also provided by adapting Altuğ's original approach. The proof proceeds in three stages: First, we estimate the contributions from the non-elliptic parts of the trace formula. Then, we apply the main result from our the previous work to isolate the $1$-dimensional representations within the elliptic part. Finally, we employ technical analytic estimates to bound the remainder terms in the elliptic part. Subjects: Number Theory (math.NT) ; Representation Theory (math.RT) MSC classes: 11F30, 11F72 Cite as: arXiv:2507.09655 [math.NT] (or arXiv:2507.09655v3 [math.NT] for this version) https://doi.org/10.48550/arXiv.2507.09655 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Yuhao Cheng [ view email ] [v1] Sun, 13 Jul 2025 14:40:56 UTC (39 KB) [v2] Tue, 17 Feb 2026 14:16:34 UTC (49 KB) [v3] Sat, 15 Aug 2026 04:04:58 UTC (41 KB) Full-text links: Access Paper: View a PDF of the paper titled Beyond endoscopy for $\mathsf{GL}_2$ over $\mathbb{Q}$ with ramification 2: bounds towards the Ramanujan conjecture, by Yuhao Cheng View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2025-07 Change to browse by: math math.RT 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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