Initial layers of Zp-extensions and Greenberg's conjecture

Abstract:

Let p be a prime number, K a finite abelian extension of Q containing p-th roots of unity and K _n the n-th layer of the cyclotomic Z _p -extension of K. Under some conditions we construct an element of K _n from an ideal class of the maximal real subfield of K _n . We determine whether its p-th root is contained by some Z _p -extension of K _n or not for each n, using the zero of p-adic L-function and the order of the ideal class group of the maximal real subfield of K _m for sufficiently large m.