This volume brings together research and survey articles in algebraic number theory and arithmetic geometry. All of them share as a general theme the mysterious connections between special values of \(L\)-functions and algebraic invariants of Galois representations. These relationships are explored primarily through the lenses of Iwasawa theory and other Galois-equivariant points of view.

The topics covered include the Galois module structure of ideal class groups, reciprocity laws in Iwasawa theory, Euler systems, \(p\)-adic \(L\)-functions, and étale cohomology—each of which has had remarkable importance in the study of \(p\)-adic Galois representations over the last few decades. In addition, the final chapters of this volume serve as an introduction to the emerging subject of special \(L\)-values in positive characteristic. This is a new direction in the general area of global function field arithmetic that is concerned with the invariants of Galois representations valued in positive characteristic, as provided by Drinfeld modules or \(t\)-modules.

Serving as the proceedings of an international conference held at ICMAT (Madrid) in May 2023, this volume is a useful resource for important techniques and approaches, as well as a source of concrete results and bibliographic references. It is of interest both to established researchers and to graduate students interested in algebraic number theory or in arithmetic geometry.