Study of Iwasawa Theory for Cyclotomic Fields.

在1997 - 1998年期间，我得到了关于实切环（阿贝尔）场的理想类群的几个结果。在这里，我总结一下主要的几点。设1为素数，k为虚阿贝尔域（满足一定条件），k_*/k为分环z_l扩展。利用复共轭作用，将k_*理想类群的l-部A_*分解为A_*=A_*^+ <对称> A_*^-。奇数部分A_*^-的结构是由Mazur和Wiles确定的。对于偶部A_*^+，推测A_*^+是有限群。目前，这个猜想还远未得到解决。设U为k_*在1处的半局部单位群。定义了由若干高斯和生成的U的子群G，并证明了A_*^+与商U/G具有“相同”的伽罗瓦模结构。我希望这个结果能对关于A_*^+.2的困难猜想有所启发。对于素数1，设h_l^+为实l-环切场的类数。我们推测，对于任意N，存在无穷多个具有h_*^+ > N的素数1，但这还没有得到证明。我证明了一个类似于这个猜想的函数场成立。

During 1997 - 1998, I obtained several results on the ideal class groups of real cyclotomic (abelian) fields. Here, I summarize the main ones.1. Let 1 be a prime number, k an imaginary abelian fields (satisfying some conditions), and k_*/k the cyclotomic Z_l-extension. The l-part A_* of the ideal class group of k_* is decomposed as A_*=A_*^+ <symmetry> A_*^- by the action of complex conjugation. The structure of the odd part A_*^- had been determined by Mazur and Wiles. As for the even part A_*^+, it is conjectured that A_*^+ is a finite group. At present, this conjecture is far to be solved. Let U be the group of semi-local units at 1 of k_*. I defined a subgroup G of U generated by certain Gauss sums, and proved that A_*^+ and the quotient U/G have the "same" Galois module structure. I hope that this result sheds some light on the difficult conjecture on A_*^+.2. For a prime number 1, let h_l^+ be class number of the real l-cyclotomic field. It is conjectured that for any N, there exist infinitely many primes 1 with h_*^+ > N.But, this is not yet proved to be true. I proved that a function field analogue of this conjecture holds.