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

Alexander Stolin

Arxiv ID: 1003.1871•Last updated: 9/13/2021

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.

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