Modular elliptic curves and hyperbolic uniformization

[2607.03830] Modular elliptic curves and hyperbolic uniformization Skip to main content Search Submit Donate Log in Mathematics > Number Theory arXiv:2607.03830 (math) [Submitted on 4 Jul 2026 ( v1 ), last revised 27 Aug 2026 (this version, v2)] Title: Modular elliptic curves and hyperbolic uniformization Authors: Neil Dummigan , Devendra Tiwari View PDF HTML (experimental) Abstract: In an article published a few years before the modularity of elliptic curves over $\Q$ was proved, Mazur \cite{maz} looked at modularity as a purely complex analytic phenomenon, defining a notion of an elliptic curve over $\Q$ having a hyperbolic uniformisation of arithmetic type. Such an elliptic curve (of conductor $N$, say) is necessarily geometrically modular, i.e. a quotient of the jacobian of the modular curve $X_0(N)$, by a morphism defined over $\Q$. We extend these ideas to elliptic curves over totally real fields of odd degree, using Shimura curves for quaternion algebras split at all finite places and one real place. In particular, we prove that the existence of a hyperbolic uniformisation of arithmetic type would imply geometric modularity. Comments: Modular Curve, Shimura Curve, Hyperbolic Geometry, Fuchsian groups Subjects: Number Theory (math.NT) MSC classes: 11G18, 11R52, 30F35 Cite as: arXiv:2607.03830 [math.NT] (or arXiv:2607.03830v2 [math.NT] for this version) https://doi.org/10.48550/arXiv.2607.03830 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Devendra Tiwari [ view email ] [v1] Sat, 4 Jul 2026 11:33:38 UTC (16 KB) [v2] Thu, 27 Aug 2026 10:27:13 UTC (17 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2026-07 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation 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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