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Mathematics of Computation

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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
Published electronically: December 21, 2001
MathSciNet review: 1898756
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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.

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Additional Information

Jesse Ira Deutsch
Affiliation: Mathematics Department, University of Botswana, Private Bag 0022, Gaborone, Botswana

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