On Kervaire--Murthy conjecture, Bernoulli and Iwasawa numbers, and zeroes of $p$-adic $L$-function

Abstract

The aim of the present paper is to establish relations between Iwasawa and Bernoulli numbers based on some results by M. Kervaire and M. P. Murthy about the structure of the $K_0$ groups of the integer group rings of cyclic groups of prime power order $p^n .$ In particular, we will prove that $\lambda_{i}\leq p-1$ under assumption that the generalized Bernoulli number $B_{1,\omega^{-i}}$ is not divisible by $p^2$. Here $\omega$ is the Teichm\"{u}ller character of $\mathbb{Z}/(p-1)\mathbb{Z}$. $\lambda_{i}=1$ if $B_{1,\omega^{-i}}$ is divisible by $p^2$. We will prove that $S_{n,i}\cong \mathbb{Z}/(p^{n+k_i})$, where $S_n$ is the Sylow $p$-subgroup of the class group of the field $\mathbb{Q}(\zeta_n)$. Here, $\zeta_n$ is a primitive $p^{n+1}$-root of unity, $\varepsilon_{i}$ are idempotents in the group ring ${\mathbb Z}_{p}[{\rm Gal}(\mathbb{Q} (\zeta_0) /\mathbb{Q})]$, $S_{n,i}=\varepsilon_i (S_n)$, and $k_i$ is the $p$-adic valuation of $B_{1,\omega^{-i}}$. At the end we will prove that $k_i \leq 1$ and also $v_p (L_p (0, \omega^j))\leq 1$ for even $j$ under certain conditions on zeroes of $L_p (0, \omega^j) .$ Throughout the paper we assume that $p$ satisfies Vandiver's conjecture.