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On some conjectures on the arithmetic of elliptic curves : 타원 곡선의 수론에 관한 몇 가지 가설들

DC Field Value Language
dc.contributor.advisor변동호-
dc.contributor.author김태경-
dc.date.accessioned2017-07-14T00:42:22Z-
dc.date.available2017-07-14T00:42:22Z-
dc.date.issued2016-02-
dc.identifier.other000000133283-
dc.identifier.urihttps://hdl.handle.net/10371/121310-
dc.description학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2016. 2. 변동호.-
dc.description.abstractThe goal of the present thesis is twofold-
dc.description.abstractwe show the two conjectures concerning the arithmetic of elliptic curves: the Stein–Watkins conjecture (for 5-isogenies) and the Gross--Zagier conjecture.

Essentially, Stein--Watkins conjecture tells us about the relations of optimal curves in given rational isogeny class of elliptic curves. In this thesis we show the two optimal curves differ by a 5-isogeny if and only if the isogeny class is '11a'.

The Gross--Zagier conjecture provides a theoretical evidence to the strong form of Birch and Swinnerton-Dyer conjecture. We show when elliptic curves have particular types of rational torsion subgroups, the order of the torsion subgroup divides certain arithmetic invariants attached to the curve.
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dc.description.tableofcontentsChapter 1. Introduction 1

Chapter 2. Elliptic curves 5

Chapter 3. Differing isogenies of optimal curves 45

Chapter 4. GrossZagier conjecture 59

Bibliography 127

Abstract (in Korean) 135
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dc.formatapplication/pdf-
dc.format.extent3564039 bytes-
dc.format.mediumapplication/pdf-
dc.language.isoen-
dc.publisher서울대학교 대학원-
dc.subjectelliptic curves-
dc.subjectBirch and Swinnerton-Dyer conjecture-
dc.subjectGross--Zagier theorem-
dc.subjectisogeny of elliptic curves-
dc.subject.ddc510-
dc.titleOn some conjectures on the arithmetic of elliptic curves-
dc.title.alternative타원 곡선의 수론에 관한 몇 가지 가설들-
dc.typeThesis-
dc.contributor.AlternativeAuthorTaekyung Kim-
dc.description.degreeDoctor-
dc.citation.pagesxiv, 137-
dc.contributor.affiliation자연과학대학 수리과학부-
dc.date.awarded2016-02-
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