Greenberg's conjecture for Dirichlet characters of order divisible by $p$

Abstract

Fix an odd prime number $p$. For an even Dirichlet character $\chi$, it is conjectured that the Iwasawa $\lambda$-invariant $\lambda_{p,\chi}$ related to the $\chi$-part of ideal class group is zero ([5], [2]). In this note, we show (under some assumptions) that there exist infinitely many characters $\chi$ of order divisible by $p$ for which the conjecture is true by using Kida's formula ([6]).