Regulator-Class Number Duality in Towers of Number Fields

This paper explores the intricate relationship between regulators and class numbers in the context of infinite towers of number fields, a concept central to modern algebraic number theory. We delve into the duality that emerges when these fundamental invariants are studied within $mathbb{Z}_p$-extensions and other generalizations of number fields. The analytic class number formula traditionally provides a direct link between the class number, the regulator, the discriminant, and the number of roots of unity of a single number field. However, in towers, their individual behaviors and their combined product exhibit complex asymptotic properties. We investigate how Iwasawa theory, particularly its main conjecture, offers a framework to understand the $p$-adic behavior of class numbers and, by extension, implies deep connections with $p$-adic regulators. The paper formalizes a "regulator-class number duality" by drawing parallels with the structures of various Iwasawa modules and their annihilators, employing tools from class field theory, Galois cohomology, and $p$-adic analysis. We present theoretical results demonstrating this duality for specific types of number field towers, highlighting how the growth of the class number ideal in the tower is balanced by the behavior of the $p$-adic regulator. The discussion includes implications for the veracity of the Iwasawa main conjecture and outlines future research directions for non-abelian extensions.