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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
DOI: https://doi.org/10.1090/S0025-5718-01-01403-X
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] H. Cohn, A Classical Invitation to Algebraic Numbers and Class Fields, Springer-Verlag, New York, 1978. MR 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] K. B. Gundlach, Die Bestimmung der Funktionen zu einigen Hilbertschen Modulgruppen, J. Reine Angew. Math. 220 (1965), 109-153. MR 33:1290
  • [5] K. B. Gundlach, Die Fixpunkte einiger Hilbertscher Modulgruppen, Math. Annalen 157 (1965), 369-390. MR 37:5153
  • [6] B. Haible, Private communication, 1997.
  • [7] F. Hirzebruch, Hilbert Modular Surfaces, Monographie No. 21 de L'Enseignement Mathématique, Geneva, Switzerland, 1973. MR 52:10750
  • [8] N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, New York, USA, 1984. MR 86c:11040
  • [9] J-P. Serre, A Course in Arithmetic, Springer-Verlag, New York, USA, 1973. MR 49:8956
  • [10] C. L. Siegel, Advanced Analytic Number Theory, Tata Institute, Bombay, India, 1980. MR 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: https://doi.org/10.1090/S0025-5718-01-01403-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

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