Arithmetic of $L$-Functions

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 a 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.