Research on ideal class groups of algebraic number fields and number theoretic functions and its applications

点击翻译按钮获取中文摘要

The purpose of this research is to develop the fundamental theory of ideal class groups of algebraic number fields further, especially taking account of Iwasawa theory. Though we did not find out a theoretical behavior of p-ambiguous ideal class groups in the first year of this research, we verified that our estimate on the density of real quadratic fields whose Iwasawa invariants for p=3 are all zero is very near to the conjectural ratio obtained by modifying Cohen-Lenstra Heuristics on class numbers. As compared with the case p>3 in which known estimates are too far from the conjectural one, our analysis is interesting. Next year, we showed that the density of real quadratic fields whose Iwasawa invariants for p=2 are all zero is zero. This is also interesting, because the density for p=3 is more than 0.7 by our previous result and the one for p>3 is conjectured to be positive. From that time to the last year of this research, we had an opportunity to have a joint research with Dr. Gen Yamamoto (Tokyo Denki University) and determined all real abelian 2-extension fields whose Iwasawa invariants for p=2 are all zero, by using genus theory and the theory of central extensions. The results obtained in this research are presented in some of conferences like as Korea-Japan Number Theory Seminar and ICM2006. Though we did not get a satisfying result about mutual application between number theory and combinatorial theory unfortunately, we played an important role in posing and solving common problems in both fields by organizing mini conferences where researchers in both fields gathered