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http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/71508完整後設資料紀錄
| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 陳其誠 | |
| dc.contributor.author | Chih-Kai Fang | en |
| dc.contributor.author | 方致凱 | zh_TW |
| dc.date.accessioned | 2021-06-17T06:02:06Z | - |
| dc.date.available | 2019-02-12 | |
| dc.date.copyright | 2019-02-12 | |
| dc.date.issued | 2019 | |
| dc.date.submitted | 2019-01-30 | |
| dc.identifier.citation | [1] J. E. Cremona, Algorithms for Modular Elliptic Curves, Cambridge University Press,1997.
[2] Atkin, A. O. L., Lehner, J. , Hecke operators on Γ0(m), Mathematische Annalen,185 no. 2(1970) 134-160. [3] Breuil, Christophe, Conrad, Brian, Diamond, Fred, Taylor, Richard, On the modularity of elliptic curves over Q: wild 3-adic exercises, Journal of the American Mathematical Society, 14 no. 4(2001), 843-939. [4] Deligne, Pierre, Formes modulaires et repr´esentations l-adiques, S´eminaire Bourbakivol. 1968/69, Expos´es 347-363, Lecture Notes in Mathematics, 179, Berlin, New York, Springer-Verlag. [5] Deligne, Pierre, La conjecture de Weil. I., Publications Math´ematiques de l’IHES, ´43(1974) 273-307. [6] Knapp, Anthony W. , Elliptic Curves Volume 40 of Mathematical notes, PrincetonUniversity Press, 1992. [7] B. Mazur, and P. Swinnerton-Dyer, Arithmetic of Weil Curves, Invent. Math.25(1974), 1-61. [8] B. Mazur, and J. Tate, and J. Teitelbaum, On p-adic analogues of the conjectures ofBirch and Swinnerton-Dyer, Invent. Math. 84 (1986), 1-48. [9] B. Mazur, and J. Tate, Refined conjectures the of Birch and Swinnerton-Dyer type,Duke Mathematical Journal, Vol.54, No. 2(1987), 711-750. [10] Ju I, Manin, Parabolic Points and Zeta-Functions of Modular Curves, Izvestiya:Mathematics, Volume 6, Issue 1(1972), 19-64. [11] J.S. Milne, Elliptic Curves, BookSurge Publishers, 2006. [12] Shimura, Goro, Introduction to the Arithmetic Theory of Automorphic Functions,Princeton University Press, 1994. [13] Silverman, Joseph H, The Arithmetic of Elliptic Curves, GTM 106, Springer-Verlag,New York, 1986. [14] Silverman, Joseph H, Advanced Topics in the Arithmetic of Elliptic Curves, GTM151, Springer-Verlag, New York, 1994. [15] Wiles, Andrew, Modular elliptic curves and Fermat’s last theorem, Annals of Mathematics, Second Series, 141, no. 3(1995), 443-551. | |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/71508 | - |
| dc.description.abstract | 橢圓曲線的狄利克雷級數在 s=1 上的階是大家所感興趣的。Mazur、Tate 和
Teitelbaum 發現一個可以計算經扭變後的橢圓曲線的狄利克雷級數在特殊點 s=1 上 是否為零點的方法。這涉及到計算模符號。本篇論文主要內容是在於整理他們所給 出的方法後，同時提出一個完整的演算法列表計算出扭變後的橢圓曲線的狄利克 雷級數在特殊點 s=1 上是否為零點。 | zh_TW |
| dc.description.abstract | The order of the Dirichlet series of the elliptic curve at s=1 is of interest to everyone.
Mazur, Tate, and Teitelbaum found a way to calculate whether the Dirichlet series of the twisted elliptic curve has a zero at the particular point s=1. This involves calculating the modulo symbol. The aim of this paper is to sort out the methods they have given, and at the same time propose a complete list of algorithms to calculate whether the Dirichlet series of the twisted elliptic curve has a zero at the special point s=1. | en |
| dc.description.provenance | Made available in DSpace on 2021-06-17T06:02:06Z (GMT). No. of bitstreams: 1 ntu-108-R03221010-1.pdf: 709909 bytes, checksum: 0802684c2e9b19f5509740ab77eaa489 (MD5) Previous issue date: 2019 | en |
| dc.description.tableofcontents | 口試委員會審定書...........................................................................................................#
誌謝................................................................................................................................... i 中文摘要.......................................................................................................................... ii ABSTRACT .................................................................................................................... iii CONTENTS .................................................................................................................... iv Chapter 1 Introduction..............................................................................................1 Chapter 2 Elliptic Curves and the Associated L-functions....................................3 2.1 Elliptic curves.................................................................................................3 2.2 The minimal Weierstrass equation..................................................................4 2.3 The reduction modulo p..................................................................................5 2.4 The conductor.................................................................................................5 2.5 The number ap ............................................................................................... 6 2.6 L-functions associated to E ............................................................................. 6 2.7 The invariant differential ................................................................................ 7 Chapter 3 Elliptic Curves and the Associated L-functions .................................... 8 3.1 The Modular Curve X0(N) ............................................................................. 8 3.2 The Index |Γ:Γ0(N)| ..................................................................................... 8 3.3 The Equivalent Classes of Cusps .................................................................. 10 3.4 The genus of X0(N) ...................................................................................... 12 3.5 Elliptic points on X0(N) ............................................................................... 12 3.6 Modular Forms ............................................................................................. 14 3.7 Hecke Operators ........................................................................................... 15 3.8 Modularity theorem ...................................................................................... 16 3.9 Mellin Transformation .................................................................................. 16 Chapter 4 Modular Symbols ................................................................................... 19 4.1 The Dual Space S2(N)∗ ............................................................................... 19 4.2 The Symbol {α,β} ......................................................................................... 22 4.3 Computing H1(X(N)(ℂ),ℤ) ........................................................................ 24 4.4 Dual Hecke Operators on H1(X(N)(ℂ),ℤ) ................................................... 26 4.5 The eigenspaces of Tp∗ .................................................................................. 28 4.6 The eigenspace of τ ....................................................................................... 29 4.7 Summary ....................................................................................................... 30 Bibliography .................................................................................................................... 32 | |
| dc.language.iso | en | |
| dc.subject | 橢圓曲線 | zh_TW |
| dc.subject | 狄利克雷級數 | zh_TW |
| dc.subject | 模形式 | zh_TW |
| dc.subject | 赫克算子 | zh_TW |
| dc.subject | 模符號 | zh_TW |
| dc.subject | elliptic curve | en |
| dc.subject | modular symbols | en |
| dc.subject | modular forms | en |
| dc.subject | Hecke operators | en |
| dc.subject | Dirichlet series | en |
| dc.title | 決定橢圓曲線上 L 函數的特殊零點 | zh_TW |
| dc.title | Determining the Special Zero of an L-function Associated to an Elliptic Curve | en |
| dc.type | Thesis | |
| dc.date.schoolyear | 107-1 | |
| dc.description.degree | 碩士 | |
| dc.contributor.oralexamcommittee | 王藹農,紀文鎮 | |
| dc.subject.keyword | 橢圓曲線,狄利克雷級數,模形式,赫克算子,模符號, | zh_TW |
| dc.subject.keyword | elliptic curve,Dirichlet series,modular forms,Hecke operators,modular symbols, | en |
| dc.relation.page | 33 | |
| dc.identifier.doi | 10.6342/NTU201900282 | |
| dc.rights.note | 有償授權 | |
| dc.date.accepted | 2019-01-31 | |
| 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 | |
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