Iwasawa theory began in the late 1950s with a series of papers by Kenkichi Iwasawa on ideal class groups in the cyclotomic tower of number fields and their relation to \(p\)-adic \( L\)-functions. The theory was later generalized by putting it in the context of elliptic curves and modular forms. The main motivation for writing this book, comprised of three volumes, was the need for a total perspective that includes the new trends of generalized Iwasawa theory. Another motivation is an update of the classical theory for class groups taking into account the changed point of view on Iwasawa theory.

The goal of this first part of the three-part publication is to explain the theory of ideal class groups, including its algebraic aspect (the Iwasawa class number formula), its analytic aspect (Leopoldt–Kubota \(L\)-functions), and the Iwasawa main conjecture, which is a bridge between the algebraic and the analytic aspects.