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Jan-Michael Feustel (1990)
Eine Klassenzahlformel für singuläre Moduln der Picardschen Modulgruppen
Masaaki Yoshida (1987)
Fuchsian Differential Equations
G. Shimura (1977)
On Abelian Varieties with Complex MultiplicationProceedings of The London Mathematical Society
H. Shiga (1988)
On the representation of the Picard modular function by t constants I-IIPublications of The Research Institute for Mathematical Sciences, 24
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The imaginary cyclic sextic fields with class numbers equal to their genus class numbersColloquium Mathematicum, 75
R. Holzapfel (2004)
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R. Holzapfel, Berlin Vladov, Sooa (2000)
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D. Mumford (1982)
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Keiji Matsumoto (1989)
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K. Koike, H. Shiga (2008)
An extended Gauss AGM and corresponding Picard modular formsJournal of Number Theory, 128
H Shiga (1988)
On the representation of the Picard modular function by $$\theta $$ θ constants I-IIPubl. RIMS Kyoto Univ., 24
G. Frey, T. Lange (2005)
Complex Multiplication
志村 五郎, 谷山 豊 (1961)
Complex multiplication of Abelian varieties and its applications to number theory
Young-Ho Park, Soun-Hi Kwon (1997)
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(1933)
Sur la théorie du corps de classes dans les corps finis et les corps loceaux
T Riedel (2007)
Arithmetic and Geometry Around Hypergeometric Functions
R. Holzapfel (1986)
Geometry and arithmetic around Euler partial differential equations
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Shimuraklassenkörper zu PicardschenModulformen
We know explicit Picard modular functions, corresponding to a family of hyperelliptic curves, with the property that their values in CM points generate abelian extensions of the associated reflex fields (Matsumoto, Ann Sc Norm Sup Pisa 16(4):557–578, 1989, Riedel, In: Arithmetic and geometry around hypergeometric functions. Birkhäuser, Basel, 2007, pp 273–285). In this note we study the number fields and their extensions occuring in this way. We show that every sextic CM field containing the fourth roots of unity is projectively generated by a singular modulus and appears as reflex field. In order to investigate the abelian extensions, we use the class field theoretic description of the field of moduli. In the unramified case we develop conditions that assure that the Picard-Shimura class field is equal to the reflex field or to the Hilbert class field. Finally, we determine these class fields for odd class numbers up to 11.
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg – Springer Journals
Published: Jun 20, 2015
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