There are two aspects (equivalent by Pontryagin duality) of the Artin reciprocity law in the classical class field theory:
(i) the representation theoretic (via the identity of \(L\)-functions) \[ \operatorname{Hom}_{\text{cont}} \bigl( \text{Gal}(\mathbb{Q}^{ab}/ \mathbb{Q}),\mathbb{C}^\times\bigr) \simeq\operatorname{Hom}_{\text{cont}} \bigl((\mathbb{A}^{(\infty)})^\times/ \mathbb{Q}^\times_+, \mathbb{C}^\times \bigr) \] (ii) geometric reciprocity \[ \text{Gal}(\mathbb{Q}^{ab}/ \mathbb{Q}) \simeq \text{GL}_1(\mathbb{A}^{(\infty)})/ \mathbb{Q}^\times_+. \] The representation theoretic aspect is generalized conjecturally by Langlands in a non-Abelian setting. Geometric reciprocity in a non-Abelian setting would be via Tannakian duality – here the Shimura variety appears as a natural object to study.
The first purpose of this book is to supply the base of the construction of the Shimura variety. The second one is to introduce integrality of automorphic forms on such varieties and hence to supply a foundation for a \(p\)-adic analytic study of automorphic forms and automorphic \(L\)-values. The mathematics discussed here is wonderful but highly nontrivial.
The book begins with a general introduction to the ideas of automorphic forms in chapter 1.
Chapter 2 discusses the classical reciprocity law, the cyclotomic reciprocity law, and the non-abelian global reciprocity law due to Shimura.
Chapter 3 recalls the generalization of the reciprocity law in the theory of elliptic modular forms. Also a brief proof of the vertical control theorem in the \(p\)-ordinary case of elliptic modular forms is included.
Chapter 4 is devoted to proving vertical control theorems, and local and global geometric reciprocity laws for Hilbert modular forms.
Chapter 5 is dedicated to proving semisimplicity of the Hecke operator action on topological nearly ordinary cohomology groups for a general reductive group.
Chapter 6, after recalling basic techniques of constructing algebro-geometric moduli schemes, contains a proof of Shimura’s global reciprocity law for Siegel modular functions.
Chapter 7 contains the construction of Shimura varieties via the integer moduli theory of abelian schemes, and the formulation of the global reciprocity law for general Shimura varieties.
Finally, Chapter 8 is dedicated to proving two important results: the vertical control theorem and irreducibility of the Igusa tower for unitary and symplectic groups. Irreducibility of the Igusa tower over the modulo \(p\) Shimura variety supplies us with the \(q\)-expansion principle – one of the most useful tools in the \(p\)-adic study of automorphic forms. Key ingredient in the proof of the vertical control theorem is the deformation theory of Serre-Tate of \(p\)-ordinary abelian schemes.
The book will certainly be useful to graduate students and researchers entering this beautiful and difficult area of research.