Torres infinitas sorprendentes. Una introducci\'on a la Teor\'ia de Iwasawa

TL;DR

This book offers a comprehensive, self-contained introduction to classical and modern Iwasawa Theory, covering algebraic number theory, $p$-adic $L$-functions, and the main conjecture, with detailed proofs and recent generalizations.

Contribution

It provides an accessible, detailed textbook in Spanish on Iwasawa Theory, including new proofs and recent extensions to elliptic curves and motives.

Findings

Explains the cyclotomic main conjecture and its implications.

Includes detailed proofs not found in standard literature.

Discusses generalizations to elliptic curves and motives.

Abstract

This text is a mostly self-contained textbook-like introduction to classical and modern Iwasawa Theory in Spanish language. It grew out of notes for a course that the authors gave at the annual conference of the Mexican Mathematical Society (SMM) in 2018. It covers some preliminaries from algebraic number theory and profinite algebra, the theory of the Iwasawa algebra and its modules, results about class groups in $\mathbb{Z}_p$-extensions, $p$-adic $L$-functions for Dirichlet characters and their various constructions (along with some complex theory of $L$-functions), the cyclotomic main conjecture in different formulations together with its most important implications, and a chapter about generalizations outlining logarithmic Iwasawa theory and generalizations of the Main conjecture to elliptic curves, motives and the ETNC. The text includes references and detailed proofs, some of which are not available in the standard literature, and is accessible for anyone with basic knowledge of algebra and some algebraic number theory.