臺灣博碩士論文加值系統

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。

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.

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

[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.