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

TL;DR

This paper explores deep connections between Iwasawa theory, Bernoulli numbers, and class groups, establishing bounds and structural results under Vandiver's conjecture and analyzing zeroes of p-adic L-functions.

Contribution

It proves new bounds on Iwasawa invariants and class group structures relating Bernoulli numbers and p-adic L-function zeroes, extending previous results by Kervaire and Murthy.

Findings

Bounded the Iwasawa lambda-invariant by p-1 under certain divisibility conditions.

Determined the structure of Sylow p-subgroups of class groups as cyclic groups with specific orders.

Established upper bounds on valuations of p-adic L-functions at zero for even characters.

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.