Computing Isogeny Volcanoes of Rank Two Drinfeld Modules

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dc.contributor.advisorGreenberg, Matthew
dc.contributor.advisorScheidler, Renate
dc.contributor.authorCaranay, Perlas
dc.contributor.committeememberBauer, Kristine
dc.contributor.committeememberBauer, Mark
dc.contributor.committeememberDimitrov, Vassil Simeonov
dc.contributor.committeememberChen, Imin
dc.date2018-06
dc.date.accessioned2018-01-25T18:45:51Z
dc.date.available2018-01-25T18:45:51Z
dc.date.issued2018-01-19
dc.description.abstractElliptic curves have long been widely studied mathematical objects. They do not only feature prominently in established areas of mathematics such as number theory, algebraic geometry, and topology, but have recently gained practical importance due to applications in coding theory and cryptography. More recently, Drinfeld modules have received increased attention due to their surprising similarity to elliptic curves - objects to which they bear little superficial resemblance. However, little is yet known about real-world applications of Drinfeld modules. Elliptic curves come in two kinds -- ordinary and supersingular. Endomorphism rings of ordinary and supersingular elliptic curves, made up of isogenies, are very different. An analogous dichotomy holds for Drinfeld modules. Recently, major progress has been achieved by researchers in explicitly computing endomorphism rings of elliptic curves using isogeny volcanoes, but very little if anything of this kind has yet been done for Drinfeld modules. Our aim here is to study the theoretical and computational aspects of isogeny volcanoes of rank two Drinfeld modules defined over finite fields and determine how to explicitly compute these mathematical structures. We establish theoretical properties of isogeny volcanoes in the Drinfeld module case. Then we design, analyze, and implement algorithms for computing (1) j-invariants and Drinfeld modular polynomials, (2) isogeny volcanoes, and (3) endomorphism rings and explicit isogenies of rank two Drinfeld modules.en_US
dc.identifier.citationCaranay, P. (2018). Computing Isogeny Volcanoes of Rank Two Drinfeld Modules (Doctoral thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca.en_US
dc.identifier.doihttp://dx.doi.org/10.11575/PRISM/5401
dc.identifier.urihttp://hdl.handle.net/1880/106320
dc.language.isoenen_US
dc.publisher.facultyScienceen_US
dc.publisher.institutionUniversity of Calgaryen
dc.rightsUniversity of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission.en_US
dc.subjectDrinfeld modulesen_US
dc.subjectisogeny volcanoesen_US
dc.subjectfunction fieldsen_US
dc.subjectalgorithmsen_US
dc.subjectelliptic curvesen_US
dc.subject.classificationEducation--Mathematicsen_US
dc.titleComputing Isogeny Volcanoes of Rank Two Drinfeld Modulesen_US
dc.typedoctoral thesisen_US
thesis.degree.disciplineMathematics & Statisticsen_US
thesis.degree.grantorUniversity of Calgaryen_US
thesis.degree.nameDoctor of Philosophy (PhD)en_US
ucalgary.item.requestcopytrue
ucalgary.thesis.checklistI confirm that I have submitted all of the required forms to Faculty of Graduate Studies.en_US
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