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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.

Ordered algebraic structures --- 512.74 --- Abelian varieties --- Modular functions --- Functions, Modular --- Elliptic functions --- Group theory --- Number theory --- Varieties, Abelian --- Geometry, Algebraic --- Algebraic groups. Abelian varieties --- 512.74 Algebraic groups. Abelian varieties --- Abelian varieties. --- Modular functions. --- Abelian extension. --- Abelian group. --- Abelian variety. --- Absolute value. --- Adele ring. --- Affine space. --- Affine variety. --- Algebraic closure. --- Algebraic equation. --- Algebraic extension. --- Algebraic number field. --- Algebraic structure. --- Algebraic variety. --- Analytic manifold. --- Automorphic function. --- Automorphism. --- Big O notation. --- CM-field. --- Characteristic polynomial. --- Class field theory. --- Coefficient. --- Complete variety. --- Complex conjugate. --- Complex multiplication. --- Complex number. --- Complex torus. --- Corollary. --- Degenerate bilinear form. --- Differential form. --- Direct product. --- Direct proof. --- Discrete valuation ring. --- Divisor. --- Eigenvalues and eigenvectors. --- Embedding. --- Endomorphism. --- Existential quantification. --- Field of fractions. --- Finite field. --- Fractional ideal. --- Function (mathematics). --- Fundamental theorem. --- Galois extension. --- Galois group. --- Galois theory. --- Generic point. --- Ground field. --- Group theory. --- Groupoid. --- Hecke character. --- Homology (mathematics). --- Homomorphism. --- Identity element. --- Integer. --- Irreducibility (mathematics). --- Irreducible representation. --- Lie group. --- Linear combination. --- Linear subspace. --- Local ring. --- Modular form. --- Natural number. --- Number theory. --- Polynomial. --- Prime factor. --- Prime ideal. --- Projective space. --- Projective variety. --- Rational function. --- Rational mapping. --- Rational number. --- Real number. --- Residue field. --- Riemann hypothesis. --- Root of unity. --- Scientific notation. --- Semisimple algebra. --- Simple algebra. --- Singular value. --- Special case. --- Subgroup. --- Subring. --- Subset. --- Summation. --- Theorem. --- Vector space. --- Zero element.