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| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 于靖 | |
| dc.contributor.author | Shih-Yu Chen | en |
| dc.contributor.author | 陳昰宇 | zh_TW |
| dc.date.accessioned | 2021-06-15T06:44:23Z | - |
| dc.date.available | 2011-07-07 | |
| dc.date.copyright | 2011-07-07 | |
| dc.date.issued | 2011 | |
| dc.date.submitted | 2011-06-30 | |
| dc.identifier.citation | N. C. Ankeny, E. Artin and S. Chowla, The class number of real quadratic number fields, Ann. of Math. 56 , 1952.
E. Artin and H. Hasse, Die beiden Erganzungssatze zum Reziprozitdtsgesetz der l^n-ten Potenzrest im Korper der l^n-ten Einheitswurzeln, Abh. Math. Sem. Univ. Hamburg. 6 (1928), 146-162; reprinted in Hasse's Mathematische Abhandlungen, Band 1, de Gruyter, Berlin, 1975, pp. 326-342. Z. Borevich and I. Shafarevich, Number Theory, Academic Press, London, 1966. A. Frohlich, Formal groups, Lecture Notes in Math. 74, Springer-Verlag, Berlin and Nwe York, 1968. I. B. Fesenko, Explicit constructions in local class field theory. Thesis, Leningrad. Univ., Leningrad, 1987. I. B. Fesenko and S .V. Vostokov, The Hilbert symbol for Lubin-Tate formal groups. II, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. 132, 1983; English transl. in J. Soviet Math. 30, 1985. I. B. Fesenko and S .V. Vostokov, Local Fields and Their Extensions, 2nd, AMS, 1993. H. Hasse, Bericht uber neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkorper. Teil II: Reziprozitatsgesetz, Jber. der DMV 6 , 1930. H. Hasse, Die Gruppe der pn-primaren Zahlen fur einen Primteiler P von p., J. Reine Angew. Math. 176(1936), 174-183. K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, 2nd, Springer-Verlag, New York, 1990. K. Iwasawa, On explicit formulas for the norm residue symbol, J. Math. Soc. Japan 20 (1968), 151-165. K. Iwasawa, Local Class Field Theory, Oxford mathematical monographs, 1986. E. Kummer, A Uber die allgemeinen ReziprozitAatsgesetze der Potenzreste. J. Reine Angew. Math. 56 (1858), 270-279. J. Lubin and J. Tate, Formal complex multiplication in local fields, Ann. of Math. 81, 1965. S. Lang, Cyclotomic fields I-II, Springer-Verlag, Berlin and New York, 1990. J. Neukirch, Class Field Theory, Springer-Verlag, Berlin and New York, 1986. I. R. Shafarevich, A general reciprocity law, Mat. Sb. 26, 1950; English transl. in Amer. Math. Soc. Transl. 4, 1956. S. V. Vostokov, Explicit form of the law of reciprocity, Izv. Akad. Nauk SSSR Ser. Mat. 42, 1978; English transl. in Math. USSR-Izv. 13, 1979. S. V. Vostokov, A norm pairing in formal modules, Izv. Akad. Nauk SSSR Ser. Mat. 43, 1979; English transl. in Math. USSR-Izv. 15, 1980. S. V. Vostokov, Symbols on formal groups, Izv. Akad. Nauk SSSR Ser. Mat. 45, 1981; English transl. in Math. USSR-Izv. 19, 1982. S. V. Vostokov, The Hilbert symbol for Lubin-Tate formal groups. I, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. 114, 1982; English transl. in J. Soviet Math. 27, 1984. S. V. Vostokov, Artin-Hasse exponentials and Bernoulli numbers, Trudy Sankt-Peterb. Mat. Obschestva 3, 1995; English transl. in Amer. Math. Soc. Transl. Ser. 2, 1995. S. V. Vostokov and V. A. Letsko, A canonical decomposition in the group of points of a Lubin-Tate formal group, J. Sov. Math 24, 1984. A. J. van der Poorten, H. J. J. Te Riele and H. C. Williams, Computer Verification of the Ankeny-Artin-Chowla Conjecture for All Primes Less Than 100 000 000 000, Mathematics of Computation, Vol. 70, No. 235 (Jul., 2001), pp. 1311-1328. A. Wiles, Higher explicit reciprocity laws, Ann. of Math. (2) 107 (1978), 235-254. L. Washington, Introduction to Cyclotomic Fields, Springer-Verlag, New York, 1982. J. Yu and J.K. Yu, A note on a geometric analogue of Ankeny-Artin-Chowla's conjecture, Contemporary Mathematics, Volume 210, 1998. | |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/48010 | - |
| dc.description.abstract | 這篇論文的第一部分是整理 I.B.Fesenko, S.V.Vostokov 以及 A.Wiles 在形式群上的希爾伯特符號之公式的工作。論文的第二部分是 Kummer 公式的幾個應用，其中包括了 Von Staudt congruence, Kummer's lemma, 以及 Ankeny-Artin-Chowla congruence。 | zh_TW |
| dc.description.abstract | This paper is a survey on explicit formulas for the Hilbert symbol on Lubin-Tate formal groups due to I.B.Fesenko, S.V.Vostokov and A.Wiles. I also give applications of Kummer's formula to Von Staudt congruence, Kummer's lemma, and Ankeny-Artin-Chowla congruence. | en |
| dc.description.provenance | Made available in DSpace on 2021-06-15T06:44:23Z (GMT). No. of bitstreams: 1 ntu-100-R98221018-1.pdf: 707450 bytes, checksum: 5b182255ff6f0e9985d74f3348e6728d (MD5) Previous issue date: 2011 | en |
| dc.description.tableofcontents | Abstract (in Chinese) i
Abstract (in English) ii Introduction 2 1 Hilbert Symbol on Lubin-Tate Formal Groups 5 1.1 The Classical Hilber Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Lubin-Tate Formal Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 The Hilbert Symbol on Lubin-Tate Formal Groups . . . . . . . . . . . . . . 8 2 The First Branch of the Explicit Formulas 11 2.1 The Ring of Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Generalized Artin-Hasse-Shafarevich Maps . . . . . . . . . . . . . . . . . . . 14 2.3 Series Associated to Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4 Primary Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.5 Decomposition of Elements in the Group of Points . . . . . . . . . . . . . . 21 2.6 The Pairing < П,. >п. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.7 The Pairing < .,. >п. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.8 Explicit Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3 The Second Branch of the Explicit Formulas 43 3.1 The Mapδ n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 The Pairing < .,. >F, m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.3 A Key Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.4 Explicit Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.5 Application to the Artin-Hasse Formulas . . . . . . . . . . . . . . . . . . . . 59 4 Applications of Kummer's Formula 61 4.1 Kummer's Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.2 A Classical Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.3 Kummer's Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.4 The Ankeny-Artin-Chowla Congruence. . . . . . . . . . . . . . . . . . . . . 74 Bibliography 80 | |
| dc.language.iso | zh-TW | |
| dc.title | 形式群上的希爾伯特符號之公式及其應用 | zh_TW |
| dc.title | Explicit Formulas for the Hilbert Symbol on Lubin-Tate Formal Groups and its Applications | en |
| dc.type | Thesis | |
| dc.date.schoolyear | 99-2 | |
| dc.description.degree | 碩士 | |
| dc.contributor.oralexamcommittee | 李華介,謝銘倫 | |
| dc.subject.keyword | 希爾伯特符號,形式群, | zh_TW |
| dc.subject.keyword | Hilbert symbol,Lubin-Tate formal groups, | en |
| dc.relation.page | 82 | |
| dc.rights.note | 有償授權 | |
| dc.date.accepted | 2011-06-30 | |
| 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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