S-Space College of Natural Sciences (자연과학대학) Dept. of Mathematical Sciences (수리과학부) Theses (Ph.D. / Sc.D._수리과학부)
On some conjectures on the arithmetic of elliptic curves
타원 곡선의 수론에 관한 몇 가지 가설들
- 자연과학대학 수리과학부
- Issue Date
- 서울대학교 대학원
- elliptic curves; Birch and Swinnerton-Dyer conjecture; Gross--Zagier theorem; isogeny of elliptic curves
- 학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2016. 2. 변동호.
- 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.