The <i>p</i>-adic limits of class numbers in ${\mathbb{Z}}_p$-towers

Abstract This paper discusses variants of Weber’s class number problem in the spirit of arithmetic topology to connect the results of Sinnott–Kisilevsky and Kionke. Let p be a prime number. We first prove the p -adic convergence of class numbers in a ${\mathbb{Z}_{p}}$ -extension of a global field and a similar result in a ${\mathbb{Z}_{p}}$ -cover of a compact 3-manifold. Secondly, we establish an explicit formula for the p -adic limit of the p -power-th cyclic resultants of a polynomial using roots of unity of orders prime to p , the p -adic logarithm, and the Iwasawa invariants. Finally, we give thorough investigations of torus knots, twist knots, and elliptic curves; we complete the list of the cases with p -adic limits being in ${\mathbb{Z}}$ and find the cases such that the base p -class numbers are small and $\nu$ ’s are arbitrarily large.