Persistent Link:
http://hdl.handle.net/10150/556600
Title:
Modular Symbols Modulo Eisenstein Ideals for Bianchi Spaces
Author:
Powell, Kevin James
Issue Date:
2015
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
The goal of this thesis is two-fold. First, it gives an efficient method for calculating the action of Hecke operators in terms of "Manin" symbols, otherwise known as "M-symbols," in the first homology group of Bianchi spaces. Second, it presents data that may be used to understand and better state an unpublished conjecture of Fukaya, Kato, and Sharifi concerning the structure of Bianchi Spaces modulo Eisenstein ideals [5]. Swan, Cremona, and others have studied the homology of Bianchi spaces characterized as certain quotients of hyperbolic 3-space [3], [13]. The first homology groups are generated both by modular symbols and a certain subset of them: the Manin symbols. This is completely analogous to the study of the homology of modular curves. For modular curves, Merel developed a technique for calculating the action of Hecke operators completely in terms of "Manin" symbols [10]. For Bianchi spaces, Bygott and Lingham outlined methods for calculating the action of Hecke operators in terms of modular symbols [2], [9]. This thesis generalizes the work of Merel to Bianchi spaces. The relevant Bianchi spaces are characterized by imaginary quadratic fields K. The methods described in this thesis deal primarily with the case that the ring of integers of K is a PID. Let p be an odd prime that is split in K. The calculations give the F_p-dimension of the homology modulo both p and an Eisenstein ideal. Data is given for primes less than 50 and the five Euclidean imaginary quadratic fields Q(√-1), Q(√-2), Q(√-3), Q(√-7), and Q(√-11). All of the data presented in this thesis comes from computations done using the computer algebra package Magma.
Type:
text; Electronic Dissertation
Keywords:
Eisenstein ideal; Hecke operator; homology; manin symbol; modular symbol; Mathematics; Bianchi
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Graduate College; Mathematics
Degree Grantor:
University of Arizona
Advisor:
Sharifi, Romyar

Full metadata record

DC FieldValue Language
dc.language.isoen_USen
dc.titleModular Symbols Modulo Eisenstein Ideals for Bianchi Spacesen_US
dc.creatorPowell, Kevin Jamesen
dc.contributor.authorPowell, Kevin Jamesen
dc.date.issued2015en
dc.publisherThe University of Arizona.en
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en
dc.description.abstractThe goal of this thesis is two-fold. First, it gives an efficient method for calculating the action of Hecke operators in terms of "Manin" symbols, otherwise known as "M-symbols," in the first homology group of Bianchi spaces. Second, it presents data that may be used to understand and better state an unpublished conjecture of Fukaya, Kato, and Sharifi concerning the structure of Bianchi Spaces modulo Eisenstein ideals [5]. Swan, Cremona, and others have studied the homology of Bianchi spaces characterized as certain quotients of hyperbolic 3-space [3], [13]. The first homology groups are generated both by modular symbols and a certain subset of them: the Manin symbols. This is completely analogous to the study of the homology of modular curves. For modular curves, Merel developed a technique for calculating the action of Hecke operators completely in terms of "Manin" symbols [10]. For Bianchi spaces, Bygott and Lingham outlined methods for calculating the action of Hecke operators in terms of modular symbols [2], [9]. This thesis generalizes the work of Merel to Bianchi spaces. The relevant Bianchi spaces are characterized by imaginary quadratic fields K. The methods described in this thesis deal primarily with the case that the ring of integers of K is a PID. Let p be an odd prime that is split in K. The calculations give the F_p-dimension of the homology modulo both p and an Eisenstein ideal. Data is given for primes less than 50 and the five Euclidean imaginary quadratic fields Q(√-1), Q(√-2), Q(√-3), Q(√-7), and Q(√-11). All of the data presented in this thesis comes from computations done using the computer algebra package Magma.en
dc.typetexten
dc.typeElectronic Dissertationen
dc.subjectEisenstein idealen
dc.subjectHecke operatoren
dc.subjecthomologyen
dc.subjectmanin symbolen
dc.subjectmodular symbolen
dc.subjectMathematicsen
dc.subjectBianchien
thesis.degree.namePh.D.en
thesis.degree.leveldoctoralen
thesis.degree.disciplineGraduate Collegeen
thesis.degree.disciplineMathematicsen
thesis.degree.grantorUniversity of Arizonaen
dc.contributor.advisorSharifi, Romyaren
dc.contributor.committeememberSharifi, Romyaren
dc.contributor.committeememberSavitt, Daviden
dc.contributor.committeememberCais, Brydenen
dc.contributor.committeememberJoshi, Kirtien
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