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Trace Classes and Quadratic Forms in the Modular Group

Published online by Cambridge University Press:  20 November 2018

Benjamin Fine
Benjamin Fine*
Affiliation:
Department of Mathematics, Fairfield University Fairfield, Connecticut 06430 U.S.A
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Abstract

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The Modular Group M is PSL2(Z) the group of linear fractional transformations with integral entries and determinant one. M has been of great interest in many diverse fields of Mathematics, including Number Theory, Automorphic Function Theory and Group Theory. In this paper we give an effective algorithm to determine, for each integer d, a complete set of representatives for the trace classes in trace d. This algorithm depends on the combinatorial group theoretic structure of M. It has been subsequently extended by Sheingorn to the general Hecke groups. The number h(d) of trace classes in trace d is equal to the ideal class number of the field The algorithm mentioned above then provides a new straightforward computational procedure for determining h(d). Finally as an outgrowth of the algorithm we present a wide generalization of the Fermat Two-Square theorem. This last result can also be derived from classical work of Gauss.

Keywords

11F0620H0511E08modular grouptrace classquadratic form
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 37 , Issue 2 , 01 June 1994 , pp. 202 - 212
Copyright
Copyright © Canadian Mathematical Society 1994

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