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

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Authors
김태경
Advisor
변동호
Major
자연과학대학 수리과학부
Issue Date
2016-02
Publisher
서울대학교 대학원
Keywords
elliptic curvesBirch and Swinnerton-Dyer conjectureGross--Zagier theoremisogeny of elliptic curves
Description
학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2016. 2. 변동호.
Abstract
The goal of the present thesis is twofold
we 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.
Language
English
URI
https://hdl.handle.net/10371/121310
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College of Natural Sciences (자연과학대학)Dept. of Mathematical Sciences (수리과학부)Theses (Ph.D. / Sc.D._수리과학부)
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