Mahler measure of the Horie unit and Weber's Class Number Problem in the Cyclotomic Zr-extension of Q

摘要

Let p be a prime number. It is an interesting problem to consider whether a prime number ldivides the class numbers of the intermediate fields of the cyclotomic Zr-extension of Q. In the case p = 2, R. Okazaki developed a theory for this problem by using Mahler measure. In this paper, we focus on the case p = 3 and show that a prime number l does not divide the class numbers of the intermediate fields of the cyclotomic Z_3-extension of Q if l satisfies l ≠1 mod 27.