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

英文摘要：

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 $λ_{i}\leq p-1$ under assumption that the generalized Bernoulli number $B_{1,ω^{-i}}$ is not divisible by $p^2$. Here $ω$ is the Teichmüller character of $\mathbb{Z}/(p-1)\mathbb{Z}$. $λ_{i}=1$ if $B_{1,ω^{-i}}$ 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 $\mathbb{Q}(ζ_n)$. Here, $ζ_n$ is a primitive $p^{n+1}$-root of unity, $\varepsilon_{i}$ are idempotents in the group ring ${\mathbb Z}_{p}[{\rm Gal}(\mathbb{Q} (ζ_0) /\mathbb{Q})]$, $S_{n,i}=\varepsilon_i (S_n)$, and $k_i$ is the $p$-adic valuation of $B_{1,ω^{-i}}$. At the end we will prove that $k_i \leq 1$ and also $v_p (L_p (0, ω^j))\leq 1$ for even $j$ under certain conditions on zeroes of $L_p (0, ω^j) .$ Throughout the paper we assume that $p$ satisfies Vandiver's conjecture.