In this thesis, we are concerned with the arithmetic of elliptic curves defined over a number field or a finite field. The main purpose of this thesis is on the orders of the reductions of a given point in the Mordell-Weil groups of elliptic curves.
In Chapter 1, we introduce elliptic curves. To begin with, we explain some properties of the Mordell-Weil groups and present important theorems on elliptic curves defined over a number field. In addition, we define division polynomials of elliptic curves, which plays an important role in Chapter 2 and 3.
In Chapter 2, we are concerned with supersingular elliptic curves. Let E be an elliptic curve defined over a finite field $F_p$ for a prime p≠2,3. Then we get the complete description of the $p^k$-th division polynomials for any positive integer k when E is supersingular. Also, we get a property of the division polynomials when E is ordinary. At last, we get a sufficient condition for ordinary primes.
In Chapter 3, we prove the elliptic analogue of Bang``s theorem. Consider an elliptic curve E over a number field K. For any non-torsion point M∈E(K), the order of the reductions M mod p runs through all but finitely many positive integers as p runs through all good primes. Moreover, it runs through all positive integers for all but finitely many point M∈E(K). For singular primes, we estimate discrete valuations of division polynomials at each singular prime when evaluated at a point.

- Advisors
- Hahn, Sang-Geun
*researcher*; 한상근*researcher*

- Description
- 한국과학기술원 : 수학과,

- Publisher
- 한국과학기술원

- Issue Date
- 1997

- Identifier
- 112602/325007 / 000935843

- Language
- eng

- Description
학위논문(박사) - 한국과학기술원 : 수학과, 1997.2, [ [ii], 48 p. ]

- Keywords
뱅의 정리; 수퍼싱귤러; 타원곡선; Division polynomial; Bang``s theorem; Supersingular; Elliptic curve; 분할다항식

- Appears in Collection
- MA-Theses_Ph.D.(박사논문)

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