請用此 Handle URI 來引用此文件：
http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/60541完整後設資料紀錄
| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 陳其誠(Ki-Seng Tan) | |
| dc.contributor.author | Yen-Sheng Wang | en |
| dc.contributor.author | 王彥勝 | zh_TW |
| dc.date.accessioned | 2021-06-16T10:21:04Z | - |
| dc.date.available | 2013-08-26 | |
| dc.date.copyright | 2013-08-26 | |
| dc.date.issued | 2013 | |
| dc.date.submitted | 2013-08-16 | |
| dc.identifier.citation | [Can] Luca Candelori, Modular Curves and Mazur's Theorem, Bacheler Thesis, 2008, Harvard
University. [Kub] D. S. Kubert, Universal bounds on the torsion of elliptic curves, Proc. London Math. Soc. 33 (1976), 193-237. [Maz] B. Mazur, Modular curves and the Eisenstein ideal, Publications Math ematiques de L'IH E S Volume 47, Number 1(1977), 33-186. [Pau] Paulo Ribenboim, Classical Theory of Algebraic Numbers, Springer, New York, 2001. [RiS] Kenneth A. Ribet and William A. Stein, Letures on Modular Forms and Hecke Operators, Manuscript, December 2011. [Sch] Alexander B. Schwartz, Elliptic Curves, Group Schemes, and Mazur's Theorem, Bacheler Thesis, 2004, Harvard University. [Sil] Joseph H. Silverman, The arithmetic of Elliptic Curves, Springer, 1986. [Was] Larry Washington, Introduction to Cyclotomis Fields, Springer, 1982. | |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/60541 | - |
| dc.description.abstract | 根據Mordell-Weil定理，橢圓曲線群在整數體是有限生成，因此其耦合子群是個有限子群。1977年，Mazur教授給了在有理體上一個很漂亮的結果，他決定了所有的耦合子群之種類。這篇論文就是要探討這漂亮定理的證明以及Mordell-Weil定理。 | zh_TW |
| dc.description.abstract | Let K be a number eld and E=K be an elliptic curve, that is, a smooth projective
curve of genus 1 with an distinguished K-rational point chosen. By the Mordell-Weil Theorem, the group of points E(K) is a nitely generated abelian group. Its structure is of the form: E(K) = Etors(K) Zr According to this theorem, we know that Etors(K) is a nite group. In 1977, Mazur [Maz] proved a beautiful theorem for K = Q. It determines all the possible torsion structures of Etors(Q). In this thesis, we try to survey on the proof of this tremendous theorem as well as that of Mordell-Weil Theorem. | en |
| dc.description.provenance | Made available in DSpace on 2021-06-16T10:21:04Z (GMT). No. of bitstreams: 1 ntu-102-R99221030-1.pdf: 727778 bytes, checksum: 808fcba2083ea1c602ac4c51d4be4e2d (MD5) Previous issue date: 2013 | en |
| dc.description.tableofcontents | Contents
Acknowledgements i Abstract (in Chinese) ii Abstract (in English) iii Contents iv List of Tables 1 1 Introduction and Notation 1 1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Preliminary 3 2.1 Weil-pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Galois Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 The Ideal Class Group and the Group of Units . . . . . . . . . . . . . . . . . . . 6 2.4 The Local Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4.1 The reduction curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4.2 The reduction map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4.3 The group E=E0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4.4 The m-torsion points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Mordell-Weil Theorem 11 3.1 Height Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 The Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.3 The Weak Mordell-Weil Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.3.1 The Kummer Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3.2 The niteness Proposition . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.3.3 The proof of the weak Mordel-Weil theorem . . . . . . . . . . . . . . . . 15 4 Mazur's Theorem 16 4.1 The Associated Galois Representation . . . . . . . . . . . . . . . . . . . . . . . . 16 4.2 Two Main Ingredients in the Proof of Theorem 2 . . . . . . . . . . . . . . . . . 17 4.3 The Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.4 The proof of Proposition 4.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 References 22 | |
| dc.language.iso | en | |
| dc.subject | Mazur定理 | zh_TW |
| dc.subject | Mordell-Weil定理 | zh_TW |
| dc.subject | 懷爾配對 | zh_TW |
| dc.subject | 賀布蘭德定理 | zh_TW |
| dc.subject | 不分枝 | zh_TW |
| dc.subject | Mordel-Weil Theorem | en |
| dc.subject | Weil-Pairing | en |
| dc.subject | Herbrand Theorem | en |
| dc.subject | Unramified | en |
| dc.title | 橢圓曲線在Q上的有限子群與Mazur定理 | zh_TW |
| dc.title | A Survey on Q-torsion group of elliptic curve and Mazur's Theorem | en |
| dc.type | Thesis | |
| dc.date.schoolyear | 101-2 | |
| dc.description.degree | 碩士 | |
| dc.contributor.oralexamcommittee | 黃柏嶧(Po-Yi Huang),黎景輝(King-Fai Lai) | |
| dc.subject.keyword | Mordell-Weil定理,Mazur定理,懷爾配對,賀布蘭德定理,不分枝, | zh_TW |
| dc.subject.keyword | Mordel-Weil Theorem,Weil-Pairing,Herbrand Theorem,Unramified, | en |
| dc.relation.page | 22 | |
| dc.rights.note | 有償授權 | |
| dc.date.accepted | 2013-08-16 | |
| dc.contributor.author-college | 理學院 | zh_TW |
| dc.contributor.author-dept | 數學研究所 | zh_TW |
| 顯示於系所單位： | 數學系 | |
文件中的檔案：
| 檔案 | 大小 | 格式 | |
|---|---|---|---|
| ntu-102-1.pdf 未授權公開取用 | 710.72 kB | Adobe PDF |
系統中的文件，除了特別指名其著作權條款之外，均受到著作權保護，並且保留所有的權利。