2022/10/12 by Jun Ueki, Ueki, Jun, Hyuga Yoshizaki +1
Mathematics · #11R23 #11R29 #11S05 #57M10 #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Primary 57K10 #Secondary 11G20
paper · pdf · doi:10.48550/arxiv.2210.06182
openalex publication_date 2022/10/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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 ℤp-extension of a global field and a similar result in a ℤp-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 ℤ and find the cases such that the base p-class numbers are small and ν's are arbitrarily large.