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| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 于靖 | |
| dc.contributor.author | Ting-Han Huang | en |
| dc.contributor.author | 黃庭瀚 | zh_TW |
| dc.date.accessioned | 2021-06-17T03:15:35Z | - |
| dc.date.available | 2018-08-01 | |
| dc.date.copyright | 2018-08-01 | |
| dc.date.issued | 2018 | |
| dc.date.submitted | 2018-07-06 | |
| dc.identifier.citation | [Br] R. Brauer, On the zeta-functions of algebraic number fields, American
Journal of Mathematics, vol. 69 no. 2, p. 243-250, 1947. [BS] Z. Borevich & I. Shafarevich, Number Theory, Academic Press, London, 1966. [CR] C.W. Curtis, I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Interscience, 1962. [De] P. Deligne, Formes modulaires et représentations l-adiques, Séminaire Bourbaki, vol. 1968/1969, exposé no. 355, Lecture Notes no. 179, Springer, p. 139-172, 1971. [DiaSh] Fred Diamond & Jerry Shurman, A First Course in Modular Forms, GTM 228, Springer. [DS] P. Deligne & J. P. Serre, Formes modulaires de pois 1, Ann. sci. E.N.S. 4-th ed. ser., t.7, p. 507-530, 1974. [HW] G. H. Hardy & E. M. Wright, An Introduction to the Theory of Numbers, 3rd edit., Oxford, 1954. [La1] Serge Lang, Algebra, vol. 1, GTM 211, Springer. [La2] Serge Lang, Introduction to Modular Forms, Grundlehren der mathematischen Wissenschaften 222, Springer, 1976. [Li] W. Li, Newforms and functional equations, Math. Ann. 212, p. 285-315, 1975. [Ll] R. Langlands, Base Change for GL2, Ann. of Math. Studies 108, Princeton University Press, 1985. [Mar] J. Martinet, Character theory and Artin L-functions, Algebraic Number Fields, edited by A. Frölish, Academic Press. [Mi] Toshitsune Miyake, Modular Forms, Springer. [Neu] J. Neukirch, Class Field Theory, GTM 280, Springer. [Ogg] A. P. Ogg, On the eigenvalues of Hecke operators, Math. Ann., vol. 179, p. 101-108, 1969. [Ra] R.A. Rankin, Contributions to the theory of Ramanujan’s function (n) and similar arithmetical functions I, II, Proc. Cambridge Phil. Soc., vol. 35, p. 351-372, 1939. [Se1] J-P. Serre, Modulars Forms and Galois Representations, Algebraic Number Fields, edited by A. Frölish, Academic Press. [Se2] J-P. Serre, Linear Representations of Finite Groups, GTM 42, Springer. [Se3] J-P. Serre, A Course in Arithmetic, GTM 7, Springer. [Se4] J-P. Serre, Local Fields, GTM 67, Springer. [Sh1] Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton University Press, 1971. [Sh2] Goro Shimura, Sur les intégrales attachées aux formes automorphes, Journal Math. Soc. Japan, vol. 11 no. 4, p. 291-311, 1959. [Sil] Joseph H. Silverman, Advanced Topics in the Arithmetics of Elliptic Curves, GTM 151, Springer. [Ta] J. T. Tate, Local Constant, Algebraic Number Fields, edited by A. Frölish, Academic Press. [Tu] J. Tunnel, Artin’s conjecture for representations of octahedral types, Bull. A.M.S. 5, p. 173-175, 1981. [Wi] A. Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math., 141, p. 443-551, 1995. | |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/69433 | - |
| dc.description.abstract | 本論文主要在介紹伽羅瓦表現與模形式之間的關係。主要的來源是Deligne和Serre在1974年所發表的論文“Formes modulaires de poid 1”。
在第二節我們先敘述了關於伽羅瓦表現與模形式的必要知識。在第三、第四節我們簡略地說明了Deligne和Serre的證明。補充的第一部分則給了一個Wiles的應用結果，剩下的部分則證明了許多我們在第二節所提到的敘述，讓這份論文更加完整。 | zh_TW |
| dc.description.abstract | In this thesis, I briefly give an introduction to the relation between Galois representations and modular forms, especially the modular forms of weight one. A main source is the paper “Formes modulaires de poid 1” written by Deligne and Serre in 1974.
In section 2, we first state some required knowledge of modular forms and Galois representations. In section 3 and 4, we give an outline of the original proof of Deligne and Serre and make some remarks. The first part of the appendix part contains a short introduction of Wiles’ work, which concerns the weight two case. The rest part of the appendix contains some computations and proof of several results mentioned in section 2, so that the thesis is more self-contained. | en |
| dc.description.provenance | Made available in DSpace on 2021-06-17T03:15:35Z (GMT). No. of bitstreams: 1 ntu-107-R05221008-1.pdf: 1719379 bytes, checksum: 54b63aeb18e098e1f53584d0611dcb53 (MD5) Previous issue date: 2018 | en |
| dc.description.tableofcontents | 1 Introduction 2
2 Preliminaries 2 2.1 Modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Hecke operators . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Newforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 L-functions and functional equations . . . . . . . . . . . . . . 6 2.5 Properties of eigenvalues . . . . . . . . . . . . . . . . . . . . . 7 2.6 Galois representations . . . . . . . . . . . . . . . . . . . . . . 8 3 Deligne-Serre lifting lemma 10 4 On the paper of Deligne and Serre 12 4.1 Existence of modular representations . . . . . . . . . . . . . . 12 4.2 On the bound of semi-simple subgroups of GL2(Fℓ) . . . . . . 16 4.3 Construction of the representation over C . . . . . . . . . . 19 4.4 The last part of the proof . . . . . . . . . . . . . . . . . . . . 20 4.5 A result of theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . 23 4.6 An inverse theorem . . . . . . . . . . . . . . . . . . . . . . . . 24 5 Appendix 28 5.1 An application of the lifting lemma . . . . . . . . . . . . . . . 28 5.2 The proof of proposition 2.7 . . . . . . . . . . . . . . . . . . . 32 5.3 Hecke algebras and modular forms . . . . . . . . . . . . . . . 34 5.4 Eichler-Shimura isomorphism . . . . . . . . . . . . . . . . . . 42 5.5 Finite subgroups of GL2(Fℓ) . . . . . . . . . . . . . . . . . . . 50 5.6 Artin L-functions . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.7 Functional equations of cusp forms . . . . . . . . . . . . . . . 57 5.8 Von Staudt–Clausen theorem . . . . . . . . . . . . . . . . . . 62 | |
| dc.language.iso | zh-TW | |
| dc.subject | 伽羅瓦表現 | zh_TW |
| dc.subject | 模形式 | zh_TW |
| dc.subject | 代數數論 | zh_TW |
| dc.subject | 朗蘭茲綱領 | zh_TW |
| dc.subject | 自守表現 | zh_TW |
| dc.subject | Automorphic representations | en |
| dc.subject | Algebraic number thoery | en |
| dc.subject | Modular forms | en |
| dc.subject | Galois representations | en |
| dc.subject | Langlands program | en |
| dc.subject | Automorphic representations | en |
| dc.subject | Algebraic number thoery | en |
| dc.subject | Modular forms | en |
| dc.subject | Galois representations | en |
| dc.subject | Langlands program | en |
| dc.title | 伽羅瓦表現與模形式 | zh_TW |
| dc.title | Galois Representations and Modular Forms | en |
| dc.type | Thesis | |
| dc.date.schoolyear | 106-2 | |
| dc.description.degree | 碩士 | |
| dc.contributor.oralexamcommittee | 余正道,魏福村,洪斌哲 | |
| dc.subject.keyword | 代數數論,模形式,伽羅瓦表現,朗蘭茲綱領,自守表現, | zh_TW |
| dc.subject.keyword | Algebraic number thoery,Modular forms,Galois representations,Langlands program,Automorphic representations, | en |
| dc.relation.page | 66 | |
| dc.identifier.doi | 10.6342/NTU201801016 | |
| dc.rights.note | 有償授權 | |
| dc.date.accepted | 2018-07-06 | |
| 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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