Algorithms Related to Subgroups of the Modular Group

Classifying subgroups of the modular group PSL_2{Z} is a fundamental problem with applications to modular forms, in addition to its group-theoretic interest. While a lot of research has been done on the congruence subgroups of PSL_2{Z}, very little is known about noncongruence subgroups. The purpose...

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Main Author:	Caranica, Constantin Cristian
Other Authors:	William Adkins
Format:	Others
Language:	en
Published:	LSU 2009
Subjects:	Mathematics
Online Access:	http://etd.lsu.edu/docs/available/etd-07092009-200839/

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spelling	ndltd-LSU-oai-etd.lsu.edu-etd-07092009-2008392013-01-07T22:52:22Z Algorithms Related to Subgroups of the Modular Group Caranica, Constantin Cristian Mathematics Classifying subgroups of the modular group PSL_2{Z} is a fundamental problem with applications to modular forms, in addition to its group-theoretic interest. While a lot of research has been done on the congruence subgroups of PSL_2{Z}, very little is known about noncongruence subgroups. The purpose of this thesis is to find and characterize small-index noncongruence subgroups of the modular group PSL_2{Z}. We use the concept of Farey symbol to describe the subgroups of PSL_2{Z}. The first part contains results concerning the geometry of subgroups of PSL_2{Z}. The second part describes a graph-theoretical approach to finding all subgroups of a given index. In the third part we describe two algorithms for testing the membership of a matrix to a subgroup given by a Farey symbol. As an application we find the noncongruence subgroups of PSL_2{Z} of index less than 10. William Adkins Helena Verrill Warren Waggenspack Mark Davidson Patrick Gilmer Robert Perlis LSU 2009-07-10 text application/pdf http://etd.lsu.edu/docs/available/etd-07092009-200839/ http://etd.lsu.edu/docs/available/etd-07092009-200839/ en unrestricted I hereby certify that, if appropriate, I have obtained and attached herein a written permission statement from the owner(s) of each third party copyrighted matter to be included in my thesis, dissertation, or project report, allowing distribution as specified below. I certify that the version I submitted is the same as that approved by my advisory committee. I hereby grant to LSU or its agents the non-exclusive license to archive and make accessible, under the conditions specified below and in appropriate University policies, my thesis, dissertation, or project report in whole or in part in all forms of media, now or hereafter known. I retain all other ownership rights to the copyright of the thesis, dissertation or project report. I also retain the right to use in future works (such as articles or books) all or part of this thesis, dissertation, or project report.
collection	NDLTD
language	en
format	Others
sources	NDLTD
topic	Mathematics
spellingShingle	Mathematics Caranica, Constantin Cristian Algorithms Related to Subgroups of the Modular Group
description	Classifying subgroups of the modular group PSL_2{Z} is a fundamental problem with applications to modular forms, in addition to its group-theoretic interest. While a lot of research has been done on the congruence subgroups of PSL_2{Z}, very little is known about noncongruence subgroups. The purpose of this thesis is to find and characterize small-index noncongruence subgroups of the modular group PSL_2{Z}. We use the concept of Farey symbol to describe the subgroups of PSL_2{Z}. The first part contains results concerning the geometry of subgroups of PSL_2{Z}. The second part describes a graph-theoretical approach to finding all subgroups of a given index. In the third part we describe two algorithms for testing the membership of a matrix to a subgroup given by a Farey symbol. As an application we find the noncongruence subgroups of PSL_2{Z} of index less than 10.
author2	William Adkins
author_facet	William Adkins Caranica, Constantin Cristian
author	Caranica, Constantin Cristian
author_sort	Caranica, Constantin Cristian
title	Algorithms Related to Subgroups of the Modular Group
title_short	Algorithms Related to Subgroups of the Modular Group
title_full	Algorithms Related to Subgroups of the Modular Group
title_fullStr	Algorithms Related to Subgroups of the Modular Group
title_full_unstemmed	Algorithms Related to Subgroups of the Modular Group
title_sort	algorithms related to subgroups of the modular group
publisher	LSU
publishDate	2009
url	http://etd.lsu.edu/docs/available/etd-07092009-200839/
work_keys_str_mv	AT caranicaconstantincristian algorithmsrelatedtosubgroupsofthemodulargroup
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Algorithms Related to Subgroups of the Modular Group

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