Abelian Varieties with Complex Multiplication and Modular Functions
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A01=Goro Shimura
Abelian extension
Abelian group
Abelian variety
Adele ring
Affine space
Affine variety
Algebraic closure
Algebraic equation
Algebraic extension
Algebraic number field
Algebraic structure
Algebraic variety
Analytic manifold
Author_Goro Shimura
Automorphic function
Automorphism
Big O notation
Category=PBF
Category=PBKF
Characteristic polynomial
Class field theory
Coefficient
Complex conjugate
Complex multiplication
Complex number
Complex torus
Degenerate bilinear form
Differential form
Discrete valuation ring
Divisor
Eigenvalues and eigenvectors
Endomorphism
eq_isMigrated=1
eq_isMigrated=2
eq_nobargain
Existential quantification
Field of fractions
Finite field
Function (mathematics)
Galois extension
Galois group
Galois theory
Generic point
Group theory
Groupoid
Hecke character
Homology (mathematics)
Homomorphism
Identity element
Irreducibility (mathematics)
Irreducible representation
Lie group
Linear subspace
Modular form
Natural number
Number theory
Polynomial
Prime factor
Prime ideal
Projective space
Projective variety
Rational mapping
Rational number
Residue field
Riemann hypothesis
Scientific notation
Semisimple algebra
Simple algebra
Singular value
Special case
Subgroup
Subring
Subset
Summation
Theorem
Vector space
Product details
- ISBN 9780691016566
- Weight: 510g
- Dimensions: 152 x 235mm
- Publication Date: 28 Dec 1997
- Publisher: Princeton University Press
- Publication City/Country: US
- Product Form: Hardback
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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.
Goro Shimura is Professor of Mathematics at Princeton University. He was awarded the Leroy P. Steele Prize in 1996 for lifetime achievement in mathematics by the American Mathematical Society. He is the author of Introduction to Arithmetic Theory of Automorphic Functions (Princeton).
