2004 Fiscal Year Final Research Report Summary
Research on class numbers of cyclotomic fields and properties of Bernoulli numbers
| Project/Area Number |
14540044
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | TOKYO UNIVERSITY OF SCIENCE |
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Principal Investigator |
AGOH Takashi Tokyo Univ.of Sci., Fac.of Tech.and Sci., Professor, 理工学部, 教授 (60112893)
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| Co-Investigator(Kenkyū-buntansha) |
SHOJI Toshiaki Nagoya Univ., Grad.School of Math., Professor, 大学院・多元数理研究科, 教授 (40120191)
HARA Tamio Tokyo Univ.of Sci., Fac.of Tech., Professor, 理工学部, 教授 (10120205)
KOBAYASHI Takao Tokyo Univ.of Sci., Fac.of Tech.and Sci., Professor, 理工学部, 教授 (90178319)
HOSOH Toshio Tokyo Univ.of Sci., Fac.of Tech.and Sci., Ass.Professor, 理工学部, 助教授 (30130339)
TANAKA Ryuichi Tokyo Univ.of Sci., Fac.of Tech.and Sci., Lecture, 理工学部, 講師 (10112898)
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| Project Period (FY) |
2002 – 2004
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| Keywords | cyclotomic field / class number formula / ideal class group / Inkeri matrix / Bernoulli numbers / Stirling numbers / Stickelkerger ideal / Kummer's formula |
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| Research Abstract |
(1)To elucidate the structure of ideal class group, we studied the relative class number h^-_p of the p-th cyclotomic field for an odd prime p. We first constructed special bases of the Stickelberger ideal corresponding to the Maillet and Inkeri type matrices whose determinants represent h^-_p. Next, we obtained a Lehmer type formula by making use of the resultant of the polynomial with coefficients related to entries of Inkeri's matrix and a special cyclotomic polynomial. Consequently, we could find new factors of h^-_p and deduced a certain condition for which each prime factor of h^-_p must satisfy. (2)Concerning special properties on Bernoulli numbers connected with h^-_p, we devised the Voronoi type congruence involving the Fermat-Euler quotient. This has many applications in various aspects. For example, Kummer type formula which is needed for the construction of the p-adic L-function and the famous Vandiver congruence can be given from this congruence. (3)We studied some convolutions of the Stirling numbers of the first and second kinds by combinatorial methods and deduced various new properties and recurrences for Bernoulli numbers of the higher order and of the second kind. Our results cover almost of all the known results, and so we believe that these will contribute in no small way to a further research on Bernoulli numbers.
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