Algebraic values of Schwarz triangle functions

  • We consider Schwarz maps for triangles whose angles are rather general rational multiples of pi. Under which conditions can they have algebraic values at algebraic arguments? The answer is based mainly on considerations of complex multiplication of certain Prym varieties in Jacobians of hypergeometric curves. The paper can serve as an introduction to transcendence techniques for hypergeometric functions, but contains also new results and examples.

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Metadaten
Author:Hironori Shiga, Jürgen Wolfart
URN:urn:nbn:de:hebis:30-11852
URL:http://www.math.uni-frankfurt.de/~wolfart/wolfart.html
Document Type:Report
Language:English
Date of Publication (online):2005/06/29
Year of first Publication:2005
Publishing Institution:Universitätsbibliothek Johann Christian Senckenberg
Release Date:2005/06/29
Tag:Schwarz triangel functions; algebraic values; complex multiplication; hypergeometric functions; transcendence
Note:
auch in: Technical Reports of Mathematical Sciences, Chiba University, 20 (2004), No. 5. Revised and extended version erscheint in den Proceedings der CIMPA Summer School AGAHF, Istanbul 2005.
Source:http://www.math.uni-frankfurt.de/~wolfart/wolfart.html , Technical Reports of Mathematical Sciences, Chiba University, 20 (2004), No. 5. Revised and extended version erscheint in den Proceedings der CIMPA Summer School AGAHF, Istanbul 2005.
HeBIS-PPN:432312889
Institutes:Informatik und Mathematik / Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
MSC-Classification:11-XX NUMBER THEORY / 11Jxx Diophantine approximation, transcendental number theory [See also 11K60] / 11J91 Transcendence theory of other special functions
11-XX NUMBER THEORY / 11Jxx Diophantine approximation, transcendental number theory [See also 11K60] / 11J95 Results involving abelian varieties
14-XX ALGEBRAIC GEOMETRY / 14Kxx Abelian varieties and schemes / 14K22 Complex multiplication [See also 11G15]
30-XX FUNCTIONS OF A COMPLEX VARIABLE (For analysis on manifolds, see 58-XX) / 30Cxx Geometric function theory / 30C20 Conformal mappings of special domains
33-XX SPECIAL FUNCTIONS (33-XX DEALS WITH THE PROPERTIES OF FUNCTIONS AS FUNCTIONS) (For orthogonal functions, see 42Cxx; for aspects of combinatorics see 05Axx; for number-theoretic aspects see 11-XX; for representation theory see 22Exx) / 33Cxx Hypergeometric functions / 33C05 Classical hypergeometric functions, 2F1
Licence (German):License LogoDeutsches Urheberrecht