Exploring the evolution from its 1950s origins, the work examines how Iwasawa theory expanded from ideal class groups in cyclotomic towers linked to p-adic L-functions to influential applications in elliptic curves and modular forms. It updates classical approaches with compelling fresh insights into p-adic Galois representations.
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 was the need for a total perspective of Iwasawa theory that includes the new trends of generalized Iwasawa theory. Another motivation is to update the classical theory for class groups, taking into account the changed point of view on Iwasawa theory. The goal of this third part of the three-part publication is to present additional aspects of the cyclotomic Iwasawa theory of $p$-adic Galois representations.