Reciprocity laws of various kinds play a central role in number
theory. In the easiest case, one obtains a transparent formulation
by means of roots of unity, which are special values of exponential
functions. A similar theory can be developed for special values of
elliptic or elliptic modular functions, and is called complex
multiplication of such functions. In 1900 Hilbert proposed the
generalization of these as the twelfth of his famous problems. In
this book, Goro Shimura provides the most comprehensive
generalizations of this type by stating several reciprocity laws in
terms of abelian varieties, theta functions, and modular functions
of several variables, including Siegel modular functions.
This subject is closely connected with the zeta function of an
abelian variety, which is also covered as a main theme in the book.
The third topic explored by Shimura is the various algebraic
relations among the periods of abelian integrals. The investigation
of such algebraicity is relatively new, but has attracted the
interest of increasingly many researchers. Many of the topics
discussed in this book have not been covered before. In particular,
this is the first book in which the topics of various algebraic
relations among the periods of abelian integrals, as well as the
special values of theta and Siegel modular functions, are treated
extensively.
General
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