A Note on Nonvanishing Properties on Mod $\ell$ Drichlet $L$-values and Application to K-groups

Let $F$ be a number field. Let $p$ be a prime number. Washington proved the $\ell$-part of the class numbers in cyclotomic $\mathbb{Z}_p$ extension of $F$ is bounded when $F$ is an abelian number field and $\ell\neq p$ is a prime. By class number formula, this is essentially a mod $\ell$ nonvanishing property of Drichlet L-functions at $s=0$. In \cite{Sinnot}, Sinnot gave a different proof by algebraic methods. In this article, we show that Sinnot's method can prove nonvanishing properties of Drichlet L-functions at $s=-k$, where $k\geq 0$ is an integer. Since the Lichtenbaurn conjecture which relates the Dedekind zeta functions at $s=-k$ and higher K-groups is proved for abelian number fields, we give some bounded results on non-$p$ part of higher K-groups in cyclotomic $\mathbb{Z}_p$ extensions of a real abelian number field.