Mahler measure and Weber's class number problem in the cyclotomic $\boldsymbol{Z}_p$-extension of $\boldsymbol{Q}$ for odd prime number $p$

Abstract

Let $p$ be a prime number and $n$ a non-negative integer. We denote by $h_{p, n}$ the class number of the $n$-th layer of the cyclotomic $\boldsymbol{Z}_p$-extension of $\boldsymbol{Q}$. Let $l$ be a prime number. In this paper, we assume that $p$ is odd and consider the $l$-divisibility of $h_{p,n}$. Let $f$ be the inertia degree of $l$ in the $p$-th cyclotomic field and $s$ the maximal exponent such that $p^s$ divides $l^{p-1}-1$. Set $r=\min\{n, s\}$. We define a certain explicit constant $G_{1}(p, r, f)$ in terms of the property of the residue class of $l$ modulo $p^r$. If $l$ is larger than $G_1(p, r, f)$, then the integer $h_{p, n}/h_{p, n-1}$ is coprime with $l$. Our proof refines Horie's method.