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

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.

28 pages. new results on lim in Z and large nu in v2. minor corrections in later versions. Theorem 3.2 inserted in v8. minor corrections in v9

References in corpus (4)

• Weber's class number problem and $p$-rationality in the cyclotomic $\hat{\mathbb{Z}}$-extension of $\mathbb{Q}$
• On the variation of the Frobenius in a non abelian Iwasawa tower
• Lang--Trotter Conjecture for CM Elliptic Curves
• On the $p$-adic limit of class numbers along a pro-$p$-extension

Cited by in corpus (1)

• The Iwasawa invariants of ${\mathbb{Z}}_p^d$-covers of links