The third topic explored by Shimura is the various algebraic relations among the periods of abelian integrals. This subject is closely connected with the zeta function of an abelian variety, which is also covered as a main theme in the book. 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. In 1900 Hilbert proposed the generalization of these as the twelfth of his famous problems. A similar theory can be developed for special values of elliptic or elliptic modular functions, and is called complex multiplication of such functions. In the easiest case, one obtains a transparent formulation by means of roots of unity, which are special values of exponential functions. Reciprocity laws of various kinds play a central role in number theory. In this book, Goro Shimura provides the most comprehensive generalizations.
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