## Tohoku Mathematical Journal 2013 June SECOND SERIES VOL. 65, NO. 2

 Tohoku Math. J. 65 (2013), 253-272

Title MAHLER MEASURE AND WEBER'S CLASS NUMBER PROBLEM IN THE CYCLOTOMIC \$\boldsymbol{Z}_p\$-EXTENSION OF \$\boldsymbol{Q}\$ FOR ODD PRIME NUMBER \$p\$

Author Takayuki Morisawa and Ryotaro Okazaki

(Received August 29, 2011, revised September 3, 2012)
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.

Mathematics Subject Classification. Primary 11R29; Secondary 11R06, 11R18.

Key words and phrases. Class number, \$\boldsymbol{Z}_p\$-extension, Mahler measure.