Hilbert Modular Forms and Iwasawa Theory
Haruzo Hida is the inventor of (and the main contributor to) the theory of p-ordinary modular and automorphic forms. His new book presents several new results on the arithmetic of p-ordinary Hilbert modular forms, as well as an exposition on the relevant technical background. The book has five chapters. Chapter 1 is an extended introduction, in which the main objects appear (two-dimensional pseudo-representations, Greenberg’s Selmer groups, deformation rings and adjoint L-invariants). Chapter 2 discusses the necessary background on automorphic forms on quaternionic algebras over totally real number fields (basic definitions, Hecke operators, the relation to representation theory of GL(2) over local fields, an overview of the Jacquet-Langlands correspondence and of Galois representations associated to Hecke eigenforms).
In chapter 3, which is the core of the book, the author first proves a special case of Fujiwara’s “R=T” theorem, from which he deduces several applications: an integral version of Jacquet-Langlands correspondence (the case of classical modular forms is treated earlier in chapter 2), the Iwasawa-theoretical version of “R=T” (for nearly p-ordinary Hilbert modular forms) and a formula for the adjoint of the L-invariant of a nearly p-ordinary Hilbert modular form. Chapter 4 sums up algebra-differential theory of classical and Hilbert modular forms in the spirit of the author’s earlier book, ‘p-adic automorphic forms on Shimura varieties’ (including the theory of Igusa varieties and control theorems for nearly p-ordinary Hecke algebras). Exceptional zeros of the adjoint p-adic L-function are interpreted in terms of extensions of Λ-adic automorphic representations. Chapter 5 treats deformation rings along the cyclotomic Zp-extension of a totally real number field, with applications to adjoint Selmer groups.
The author’s style, which is a mixture of expository treatment and new research, will not be to everybody’s liking. The book itself has a fair share of confusing or incorrect statements and definitions, which makes it ideal for graduate students, who will learn a lot by trying to correct the abundant inaccuracies in the text.