The l-class group of the Z_p-extension over the rational field

Abstract

Let p be an odd prime, and let B_∞ denote the Z_p-extension over the rational field. Let l be an odd prime different from p. The question whether the l-class group of B_∞ is trivial has been considered in our previous papers mainly for the case where l varies with p fixed. We give a criterion, for checking the triviality of the l-class group of B_∞, which enables us to discuss the triviality when p varies with l fixed. As a consequence, we find that, if l does not exceed 13 and p does not exceed 101, then the l-class group of B_∞ is trivial.