2004 Fiscal Year Final Research Report Summary
Refinement of Iwasawa theory and its applications
| Project/Area Number |
14340016
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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 Metropolitan University |
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Principal Investigator |
KURIHARA Masato Tokyo Metropolitan University, Faculty of Science, Associate Professor, 理学研究科, 助教授 (40211221)
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| Co-Investigator(Kenkyū-buntansha) |
MIYAKE Katsuya Tokyo Metropolitan University, Faculty of Science, Professor, 理学研究科, 教授 (20023632)
NAKAMULA Ken Tokyo Metropolitan University, Faculty of Science, Professor, 理学研究科, 教授 (80110849)
KAWASAKI Takeshi Tokyo Metropolitan University, Faculty of Science, Assistant, 理学研究科, 助手 (40301410)
MATSUNO Kazuo Tokyo Metropolitan University, Faculty of Science, Assistant, 理学研究科, 助手 (40332936)
KURATA Toshihiko Tokyo Metropolitan University, Faculty of Science, Assistant, 理学研究科, 助手 (40311899)
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| Project Period (FY) |
2002 – 2004
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| Keywords | Iwasawa theory / Iwasawa main conjecture / Iwasawa module / ideal class group / Fitting ideal |
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| Research Abstract |
The central theme of Iwasawa theory is the main conjecture which states that the characteristic ideal of an algebraic object like Iwasawa modules is essentially generated by the p-adic L-function. Namely, the characteristic polynomial of a certain Galois action and the orders of certain arithmetic objects like ideal class groups are determined by the zeta values. In this research, we could prove that the p-adic L-functions (in a wide sense) have more information than the characteristic polynomials and the orders. More precisely, we could show that the Fitting ideals which have more information than the characteristic ideals are determined by the p-adic L-functions.
First of all, we defined the Stickelberger ideal for an imaginary abelian field K in a different way from that of Iwasawa and Sinnott, and proposed a conjecture which claims that this ideal coincides with the 0-th Fitting ideal of the class group of K Our Stickelberger ideal is defined by using several Stickelberger elements of subfields and extension fields of K, so this ideal is regarded as an analytic object, and our conjecture can be regarded as a refinement of usual main conjecture. We proved this conjecture in several cases. We also proposed an analogous conjecture for the Iwasawa module of the cyclotomic Z_{p} extension of K, and proved it completely.
We also generalized a structure theorem by Kolyvagin and Rubin for abelian fields to general CM fields. We proved that the higher Fitting ideals of the class group are generated by elements constructed from Stickelberger elements. This gives a typical example of the refinement of Iwasawa theory in our sense.
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