“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 future work in the theory.”
“The prerequisities 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 representations attached to them 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 don’t provide proofs.”
Let us analyse more precisely the content of the book. In Chapter 2 the author introduces the basic idea that is central to the construction of eigenvarieties in Chapter 3. To be precise, given a family of commuting operators acting on some space or module, one attaches an algebraic object (the eigenalgebra) that parametrizes the systems of eigenvalues for those operators appearing in the said space or module. The motivating example of such construction is the action of Hecke operators on spaces of modular forms.
Chapter 3 contains a version at large of this idea: the author explains the construction of eigenvarieties from suitable families of Banach modules with action of a commutatative algebra and proves their general properties. This chapter was inspired by the article by K. Buzzard [Lond. Math. Soc. Lect. Note Ser. 320, 59–120 (2007; Zbl 1230.11054)], but the present exposition is completely self-contained.
Chapters 4, 5, and 6 are devoted to the theory of modular symbols, and their connection to modular forms. Chapter 4 introduces the abstract theory of modular symbols, including their cohomological interpretation. Chapter 5 exposes the classical theory of modular symbols as developed by Manin, Shokurov, Amice-Velu and Mazur-Tate-Teitelbaum. Chapter 6 develops the theory, initiated by Stevens, and developed by Pollack-Stevens and Bellaïche, of rigid analytic modular symbols.
Chapter 7 gives the construction of the eigencurve using families of rigid analytic modular symbols, and compares this eigencurve to the one constructed by R. Coleman and B. Mazur [Lond. Math. Soc. Lect. Note Ser. 254, 1–113 (1998; Zbl 0932.11030)] using overconvergent modular forms.
In Chapter 8 the author uses the eigencurve to extend the construction of \(p\)-adic \(L\)-function of Mazur-Tate-Teitelbaum and Stevens to most overconvergent eigenforms, including the classical eigenforms of critical slope as well as (most) classical forms of weight one. It turns out that these \(p\)-adic \(L\)-functions can be put together to define a family of \(L\)-functions on the eigencurve.
Chapter 9 gives the construction of the adjoint \(p\)-adic \(L\)-function on the eigencurve. This chapter was inspired by W. Kim’s 2006 Berkeley Thesis [Ramifications points on the eigencurve. Berkeley: University of California (PhD Thesis) (2006)].
The prerequisities for the reader are rather high. On the other hand, complete proofs (or detailed references) of all statements are given and many exercises (with their solutions or hints) are included, hence the book may be addressed to graduate students working in this beautiful area of number theory and arithmetic algebraic geometry. This is a welcome addition to the literature in a field.