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The discriminant 10 Shimura curve and its associated Heun functions
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
2016 (English)In: Bulletin of the London Mathematical Society, ISSN 0024-6093, E-ISSN 1469-2120, Vol. 48, 957-967 p.Article in journal (Refereed) Published
Abstract [en]

The Shimura curve of discriminant 10 is uniformized by a subgroup of an arithmetic $(2,2,2,3)$ quadrilateral group. We derive the differential structure of the ring of modular forms for the Shimura curve and relate the ring generators to explicit Heun functions for the quadrilateral group. We also show that the Picard–Fuchs equation of the associated family of abelian surfaces has solutions that are modular forms. These results are used to completely describe the exceptional sets of the Heun functions, and we show how to find examples like \[ Hl\left(\frac{27}{2},\frac{7}{36}; \frac{1}{12},\frac{7}{12},\frac{2}{3},\frac{1}{2}; -\frac{96}{25}\right)=\frac{2^{1/2}5^{2/3}}{3^{4/3}}. \]

Place, publisher, year, edition, pages
Oxford University Press, 2016. Vol. 48, 957-967 p.
Keyword [en]
Shimura curves, modular forms, Heun functions, Picard-Fuchs equations, exceptional sets, CM periods
National Category
Algebra and Logic
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kau:diva-47546DOI: 10.1112/blms/bdw055OAI: oai:DiVA.org:kau-47546DiVA: diva2:1058865
Available from: 2016-12-21 Created: 2016-12-21 Last updated: 2017-01-12Bibliographically approved

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Granath, Håkan
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CiteExportLink to record
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Citation style
  • apa
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  • Other locale
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