2010 Fiscal Year Final Research Report

Iwasawa theory for abelian extensions over imaginary quadratic fields and elliptic units

Research Project

Project/Area Number	19740020
Research Category	Grant-in-Aid for Young Scientists (B)
Allocation Type	Single-year Grants
Research Field	Algebra
Research Institution	Okayama University of Science
Principal Investigator	AOKI Miho Okayama University of Science, 総合理工学部, 准教授 (10381451)
Project Period (FY)	2007 – 2010
Keywords	イデアル類群 / ガウス和 / 円単数 / 楕円単数 / 代数的K群 / Brumer予想 / Coates-Sinnott予想
Research Abstract	I studied the relation between Brumer conjecture and Coates-Sinnott conjecture. The former is related to ideal class groups of number fields, and the latter is related to K-groups of the rings of integers. In my paper "A note on the Coates-Sinnott conjecture, Bulletin of the London Mathematical Society 41, 613-620, 2009", I proved that the former conjecture implies the latter conjecture under some assumptions. In another paper "On some exact sequences in the theory of cyclotomic fields, Journal of Algebra 320, 4156-4177, 2008", I showed the existence of some exact sequences which relate both the plus part and the minus part of Iwasawa modules. Furthermore, I formulated some conjecture about K-groups of the rings of integers of cyclotomic fields, and described an order of etale cohomology groups by using cyclotomic units and Gauss sums. At the end of the study, I considered general abelian number fields and abelian extensions over imaginary quadratic fields.