Weber's class number problem and its variants

TL;DR

This paper surveys Weber's class number problem and its variants, exploring their connections to units, Pell's equations, and $p$-adic class number limits, with numerical insights from knots and elliptic curves.

Contribution

It provides a comprehensive overview linking Weber's problem to arithmetic topology, units, and $p$-adic class number behavior, including new numerical investigations.

Findings

Relation between class numbers and units

$p$-adic limits of class numbers in towers

Numerical evidence from knots and elliptic curves

Abstract

We survey Weber's class number problem and its variants in the spirit of arithmetic topology; we recollect some history, present a relation to certain units and generalized Pell's equation, and overview a study of the $p$-adic limits of class numbers in $\mathbb{Z}_p$-towers together with numerical investigation for knots and elliptic curves.