Study of Iwasawa Theory for Cyclotomic Fields.

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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.