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

TL;DR

This paper investigates the $p$-adic limits of class numbers in $Z_p$-extensions of global fields and 3-manifolds, providing explicit formulas and analyzing specific cases like knots and elliptic curves.

Contribution

It establishes the $p$-adic convergence of class numbers in $Z_p$-extensions and derives explicit formulas for their limits, connecting number theory and topology.

Findings

Proves $p$-adic convergence of class numbers in $Z_p$-extensions.

Provides explicit formulas for $p$-adic limits of cyclic resultants.

Identifies cases with $p$-adic limits in $Z$ and small class numbers.

Abstract

This article 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.