[2608.14495] Shimura curves of discriminant 14 and 15 and associated Heun Functions Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Number Theory arXiv:2608.14495 (math) [Submitted on 14 Aug 2026] Title: Shimura curves of discriminant 14 and 15 and associated Heun Functions Authors: Harshavardhan Reddy , Devendra Tiwari View a PDF of the paper titled Shimura curves of discriminant 14 and 15 and associated Heun Functions, by Harshavardhan Reddy and Devendra Tiwari View PDF HTML (experimental) Abstract: The Shimura curve of discriminant $D$ for $D=14, 15$ is uniformized by a subgroup of an arithmetic quadrilateral Fuchsian group $(2, 2, 2, q)$, where $q=4, 6$. We relate the generator of the ring of quaternionic modular forms on this Shimura curve to explicit Heun functions for the quadrilateral group. We also discuss how the Picard-Fuchs equation of the associated family of abelian surfaces has solutions that are modular forms on $X^{D}(1) / W_{D}$, where $W_D$ is the full group of Atkin-Lehner involutions. This leads us to completely describe the rational exceptional sets of the associated Heun functions, and the algebraic values attained by the Heun function on these points, for example $${\rm He}\left( 81, \frac{1}{2}; \frac{1}{3}, \frac{1}{6}, \frac{1}{2}, \frac{1}{2};-\frac{729}{112}\right)= \left( \frac{2^2 \cdot 3^3 \cdot 5^3}{7^5} \right)^{\frac{1}{6}}$$. Subjects: Number Theory (math.NT) MSC classes: 11F03, 33E30, 11J91 (primary), 33E30, 11G18 (secondary) Cite as: arXiv:2608.14495 [math.NT] (or arXiv:2608.14495v1 [math.NT] for this version) https://doi.org/10.48550/arXiv.2608.14495 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Devendra Tiwari [ view email ] [v1] Fri, 14 Aug 2026 17:10:00 UTC (25 KB) Full-text links: Access Paper: View a PDF of the paper titled Shimura curves of discriminant 14 and 15 and associated Heun Functions, by Harshavardhan Reddy and Devendra Tiwari View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2026-08 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