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

Part of: Algebraic number theory: global fields Arithmetic algebraic geometry Algebraic number theory: local and $p$-adic fields Low-dimensional topology

Published online by Cambridge University Press:  23 May 2025

JUN UEKI  and
HYUGA YOSHIZAKI
JUN UEKI
Affiliation:
Department of Mathematics, Faculty of Science, Ochanomizu University; 2-1-1 Otsuka, Bunkyo-ku, 112-8610, Tokyo, Japan. e-mail:uekijun46@gmail.com
HYUGA YOSHIZAKI
Affiliation:
Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science; 2641, Yamazaki, Noda-shi, 278-8510, Chiba, Japan. e-mail:yoshizaki.hyuga@gmail.com
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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.

MSC classification

Primary: 11R23: Iwasawa theory 11R29: Class numbers, class groups, discriminants
Secondary: 11G20: Curves over finite and local fields 57M10: Covering spaces 11S05: Polynomials
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , First View , pp. 1 - 31
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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