Joël Bellaïche - The Eigenbook: Eigenvarieties, families of Galois representations, p-adic L-functions

The aim of this book is to give a gentle but complete introduction to the two interrelated theory of p-adic families of modular forms and of p-adic L-functions of modular forms. These theories and their generalizations to automorphic forms for group of higher ranks are now of fundamental importance in number theory. To our knowledge, this is the first textbook containing this material, which now is to be found scattered in many articles published during the last 40 years. Here we present a self-contained exposition of the theory leading to the construction of the eigencurves, and of the families of p-adic L-functions and adjoint p-adic L-functions on them.

The book is intended both for beginners in the field (graduate students and established researchers in other fields alike) and for researchers working in the field of p-adic families of automorphic forms and p-adic L-functions, who want a solid foundation, in one place, for further work in the theory. Actually, I wrote it in a large part for myself, to help me prepare for my current work on p-adic L-functions and p-adic families of automorphic forms, with a view toward the Beilinson-Bloch-Kato conjectures. In particular, while the focus is for families of modular forms, the main tools used are presented in an abstract way in view of their applications to higher-dimensional situation (e.g. the eigenvariety machine, or the theory of the L-ideal used to construct adjoint p-adic L-functions).

The prerequisites for this book includes a familiarity with basic commutative algebra, algebraic geometry (the language of schemes), and algebraic number theory. In addition, a basic knowledge of the theory of classical modular forms, and the Galois representation attached to them (for example, the content of a book like []) is needed, as well as the very basics in the theory of rigid analytic geometry. Even so, we take care to recall the definitions and results we need in these theories, with precise references when we dont provide proofs.

Let us describe the content of this book.

In Chap. tries to help the reader to develop an intuition of this basic idea, with the analysis of many special cases and examples.

Chapter presents a version at large of this idea, the eigenvariety machine. It is largely inspired from the very readable article by K. Buzzard of the same title. The main change is that we offer a completely self-contained presentation, proving the results of p-adic functional analysis that we need (due to Serre, Lazard, Coleman). We also give a more general and more precise construction (removing some unnecessary hypotheses, and describing a more convenient admissible covering, which will be of much help for the rest of the book), and complete it with many results due to Chenevier, including the extremely important Cheneviers comparisons theorem.

Chapters recalls the method of Stevens to construct the p-adic L-function of classical cuspidal non-critical slope modular form.