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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

A computational approach to Hilbert modular group fixed points


Author: Jesse Ira Deutsch
Journal: Math. Comp. 71 (2002), 1271-1280
MSC (2000): Primary 11-04, 11Y35; Secondary 32-04
Published electronically: December 21, 2001
MathSciNet review: 1898756
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Abstract | References | Similar Articles | Additional Information

Abstract: Some useful information is known about the fundamental domain for certain Hilbert modular groups. The six nonequivalent points with nontrivial isotropy in the fundamental domains under the action of the modular group for $\mathbf{Q} ( \sqrt 5 )$, $\mathbf{Q}( \sqrt 2 )$, and $\mathbf{Q} ( \sqrt 3 )$ have been determined previously by Gundlach. In finding these points, use was made of the exact size of the isotropy groups. Here we show that the fixed points and the isotropy groups can be found without such knowledge by use of a computer scan. We consider the cases $\mathbf{Q} ( \sqrt 5 )$ and $\mathbf{Q} ( \sqrt 2 )$. A computer algebra system and a C compiler were essential in perfoming the computations.


References [Enhancements On Off] (What's this?)

  • [1] Harvey Cohn, A classical invitation to algebraic numbers and class fields, Springer-Verlag, New York-Heidelberg, 1978. With two appendices by Olga Taussky: “Artin’s 1932 Göttingen lectures on class field theory” and “Connections between algebraic number theory and integral matrices”; Universitext. MR 506156 (80c:12001)
  • [2] J. I. Deutsch, Identities on Modular Forms in Several Variables Derivable from Hecke Transformations, Dissertation, Brown University, Providence, R.I., USA, 1986.
  • [3] F. Götzky, Über eine zahlentheoretische Anwendung von Modulfunktionen zweier Veränderlicher, Math. Ann. 100 (1928), 411-437.
  • [4] Karl-Bernhard Gundlach, Die Bestimmung der Funktionen zu einigen Hilbertschen Modulgruppen, J. Reine Angew. Math. 220 (1965), 109–153 (German). MR 0193069 (33 #1290)
  • [5] Karl-Bernhard Gundlach, Die Fixpunkte einiger Hilbertscher Modulgruppen, Math. Ann. 157 (1965), 369–390 (German). MR 0229579 (37 #5153)
  • [6] B. Haible, Private communication, 1997.
  • [7] Friedrich E. P. Hirzebruch, Hilbert modular surfaces, Secrétariat de l’Enseignement Mathématique, Université de Genève, Geneva, 1973. Série des Conférences de l’Union Mathématique Internationale, No. 4; Monographie No. 21 de l’Enseignement Mathématique. MR 0389921 (52 #10750)
  • [8] Neal Koblitz, Introduction to elliptic curves and modular forms, Graduate Texts in Mathematics, vol. 97, Springer-Verlag, New York, 1984. MR 766911 (86c:11040)
  • [9] J.-P. Serre, A course in arithmetic, Springer-Verlag, New York-Heidelberg, 1973. Translated from the French; Graduate Texts in Mathematics, No. 7. MR 0344216 (49 #8956)
  • [10] Carl Ludwig Siegel, Advanced analytic number theory, 2nd ed., Tata Institute of Fundamental Research Studies in Mathematics, vol. 9, Tata Institute of Fundamental Research, Bombay, 1980. MR 659851 (83m:10001)

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Additional Information

Jesse Ira Deutsch
Affiliation: Mathematics Department, University of Botswana, Private Bag 0022, Gaborone, Botswana
Email: deutschj_1729@yahoo.com

DOI: http://dx.doi.org/10.1090/S0025-5718-01-01403-X
PII: S 0025-5718(01)01403-X
Keywords: Modular group, fundamental domain, quadratic field
Received by editor(s): January 6, 2000
Received by editor(s) in revised form: September 7, 2000
Published electronically: December 21, 2001
Article copyright: © Copyright 2001 American Mathematical Society