The main contribution of this book is a unified understanding of two fundamental topics in number theory: periods of abelian varieties with complex multiplication and special values of \(L\)-series of number fields. The author introduces an invariant attached to an ideal class of a totally real algebraic number field and presents the construction of both the Stark-Shintani units and the Shimura period symbol from this invariant. He introduces an absolute CM-period symbol with remarkable properties.
The new invariant associated to an ideal class is introduced along a factorization process of the value at zero of the first derivative of the partial zeta function attached to the class, which also produces the absolute period symbol. A similar construction for the second derivative of the partial zeta function is outlined in one of the appendices.
In order to facilitate the comprehension of such deep links, the book contains systematic expositions of the involved topics: generalized multiple gamma functions, Stark-Shintani units, period symbols and Eisenstein series on \(GL(2)\) with a limit formula of Kronecker’s type. All the topics are covered with great detail, including every technical calculation needed for the development of the results. The author has made a remarkable effort to include many completely developed numerical examples to illustrate the text and many exercises at the end of every chapter.
The book is essentially self-contained, but it assumes a good knowledge of basic algebraic number theory. It is intended for graduate students and researchers in number theory and automorphic forms.