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The discriminant 10 Shimura curve and its associated Heun functions
Author(s) -
Baba Srinath,
Granath Håkan
Publication year - 2016
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdw055
Subject(s) - mathematics , discriminant , abelian group , pure mathematics , quadrilateral , group (periodic table) , modular form , ring (chemistry) , algebra over a field , differential (mechanical device) , arithmetic , physics , chemistry , organic chemistry , finite element method , artificial intelligence , computer science , engineering , thermodynamics , aerospace engineering
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 H l 27 2 , 7 36 ; 1 12 , 7 12 , 2 3 , 1 2 ; − 96 25 =2 1 / 25 2 / 33 4 / 3.