Greenberg's conjecture and cyclotomic towers

Abstract

1. Greenberg’s conjecture. In the late 1950’s Iwasawa introduced a powerful technique for studying class groups and unit groups of number elds. Motivated by the theory of curves over nite elds, Iwasawa’s theory of Zp-extensions has since become a widely used tool in algebraic number theory, Galois theory, and arithmetic geometry. We describe in this section a conjecture of Greenberg concerning the structure of a classical Iwasawa module, and we mention a Galois-theoretic consequence concerning free prop-extensions of number elds. Let K be an algebraic number eld and p an odd prime. By a multiple Zp-extension K1=K we mean a Galois extension with Galois group ’Z d for some positive integer d. In what follows we will be particularly interested in two such extensions of K for which we reserve the following notation: K cyc =K denotes the cyclotomic Zp-extension of K. e K=K denotes the compositum of all Zp-extensions of K. Let F be a nite extension of K contained in K1, and denote by A(F ) the Sylow p-subgroup of the ideal class group of F . The Galois group of F=K acts onA(F ) in the natural way, makingA(F ) into aZp[Gal(F=K)]-module. As F varies over all nite subextensions the A(F ) form an inverse system (under norm maps) and we denote by A the inverse limit. The group A then carries a natural structure as a module over the Iwasawa algebra