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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.

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This book provides an accessible and self-contained introduction to the theory of algebraic curves over a finite field, a subject that has been of fundamental importance to mathematics for many years and that has essential applications in areas such as finite geometry, number theory, error-correcting codes, and cryptology. Unlike other books, this one emphasizes the algebraic geometry rather than the function field approach to algebraic curves. The authors begin by developing the general theory of curves over any field, highlighting peculiarities occurring for positive characteristic and requiring of the reader only basic knowledge of algebra and geometry. The special properties that a curve over a finite field can have are then discussed. The geometrical theory of linear series is used to find estimates for the number of rational points on a curve, following the theory of Stöhr and Voloch. The approach of Hasse and Weil via zeta functions is explained, and then attention turns to more advanced results: a state-of-the-art introduction to maximal curves over finite fields is provided; a comprehensive account is given of the automorphism group of a curve; and some applications to coding theory and finite geometry are described. The book includes many examples and exercises. It is an indispensable resource for researchers and the ideal textbook for graduate students.

Curves, Algebraic. --- Finite fields (Algebra) --- Modular fields (Algebra) --- Algebra, Abstract --- Algebraic fields --- Galois theory --- Modules (Algebra) --- Algebraic curves --- Algebraic varieties --- Abelian group. --- Abelian variety. --- Affine plane. --- Affine space. --- Affine variety. --- Algebraic closure. --- Algebraic curve. --- Algebraic equation. --- Algebraic extension. --- Algebraic function. --- Algebraic geometry. --- Algebraic integer. --- Algebraic number field. --- Algebraic number theory. --- Algebraic number. --- Algebraic variety. --- Algebraically closed field. --- Applied mathematics. --- Automorphism. --- Birational invariant. --- Characteristic exponent. --- Classification theorem. --- Clifford's theorem. --- Combinatorics. --- Complex number. --- Computation. --- Cyclic group. --- Cyclotomic polynomial. --- Degeneracy (mathematics). --- Degenerate conic. --- Divisor (algebraic geometry). --- Divisor. --- Dual curve. --- Dual space. --- Elliptic curve. --- Equation. --- Fermat curve. --- Finite field. --- Finite geometry. --- Finite group. --- Formal power series. --- Function (mathematics). --- Function field. --- Fundamental theorem. --- Galois extension. --- Galois theory. --- Gauss map. --- General position. --- Generic point. --- Geometry. --- Homogeneous polynomial. --- Hurwitz's theorem. --- Hyperelliptic curve. --- Hyperplane. --- Identity matrix. --- Inequality (mathematics). --- Intersection number (graph theory). --- Intersection number. --- J-invariant. --- Line at infinity. --- Linear algebra. --- Linear map. --- Mathematical induction. --- Mathematics. --- Menelaus' theorem. --- Modular curve. --- Natural number. --- Number theory. --- Parity (mathematics). --- Permutation group. --- Plane curve. --- Point at infinity. --- Polar curve. --- Polygon. --- Polynomial. --- Power series. --- Prime number. --- Projective plane. --- Projective space. --- Quadratic transformation. --- Quadric. --- Resolution of singularities. --- Riemann hypothesis. --- Scalar multiplication. --- Scientific notation. --- Separable extension. --- Separable polynomial. --- Sign (mathematics). --- Singular point of a curve. --- Special case. --- Subgroup. --- Sylow theorems. --- System of linear equations. --- Tangent. --- Theorem. --- Transcendence degree. --- Upper and lower bounds. --- Valuation ring. --- Variable (mathematics). --- Vector space.

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This book presents a development of the basic facts about harmonic analysis on local fields and the n-dimensional vector spaces over these fields. It focuses almost exclusively on the analogy between the local field and Euclidean cases, with respect to the form of statements, the manner of proof, and the variety of applications.The force of the analogy between the local field and Euclidean cases rests in the relationship of the field structures that underlie the respective cases. A complete classification of locally compact, non-discrete fields gives us two examples of connected fields (real and complex numbers); the rest are local fields (p-adic numbers, p-series fields, and their algebraic extensions). The local fields are studied in an effort to extend knowledge of the reals and complexes as locally compact fields.The author's central aim has been to present the basic facts of Fourier analysis on local fields in an accessible form and in the same spirit as in Zygmund's Trigonometric Series (Cambridge, 1968) and in Introduction to Fourier Analysis on Euclidean Spaces by Stein and Weiss (1971).Originally published in 1975.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Fourier analysis. --- Local fields (Algebra) --- Fields, Local (Algebra) --- Algebraic fields --- Analysis, Fourier --- Mathematical analysis --- Corps algébriques --- Fourier analysis --- 511 --- 511 Number theory --- Number theory --- Local fields (Algebra). --- Harmonic analysis. Fourier analysis --- Fourier Analysis --- Abelian group. --- Absolute continuity. --- Absolute value. --- Addition. --- Additive group. --- Algebraic extension. --- Algebraic number field. --- Bessel function. --- Beta function. --- Borel measure. --- Bounded function. --- Bounded variation. --- Boundedness. --- Calculation. --- Cauchy–Riemann equations. --- Characteristic function (probability theory). --- Complex analysis. --- Conformal map. --- Continuous function. --- Convolution. --- Coprime integers. --- Corollary. --- Coset. --- Determinant. --- Dimension (vector space). --- Dimension. --- Dirichlet kernel. --- Discrete space. --- Distribution (mathematics). --- Endomorphism. --- Field of fractions. --- Finite field. --- Formal power series. --- Fourier series. --- Fourier transform. --- Gamma function. --- Gelfand. --- Haar measure. --- Haar wavelet. --- Half-space (geometry). --- Hankel transform. --- Hardy's inequality. --- Harmonic analysis. --- Harmonic function. --- Homogeneous distribution. --- Integer. --- Lebesgue integration. --- Linear combination. --- Linear difference equation. --- Linear map. --- Linear space (geometry). --- Local field. --- Lp space. --- Maximal ideal. --- Measurable function. --- Measure (mathematics). --- Mellin transform. --- Metric space. --- Modular form. --- Multiplicative group. --- Norbert Wiener. --- P-adic number. --- Poisson kernel. --- Power series. --- Prime ideal. --- Probability. --- Product metric. --- Rational number. --- Regularization (mathematics). --- Requirement. --- Ring (mathematics). --- Ring of integers. --- Scalar multiplication. --- Scientific notation. --- Sign (mathematics). --- Smoothness. --- Special case. --- Special functions. --- Subgroup. --- Subring. --- Support (mathematics). --- Theorem. --- Topological space. --- Unitary operator. --- Vector space. --- Analyse harmonique (mathématiques) --- Analyse harmonique (mathématiques) --- Corps algébriques

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Charles Favre and Thomas Gauthier present new mathematical research in the field of arithmetic dynamics. Specifically, the authors study one-dimensional algebraic families of pairs given by a polynomial with a marked point. Combining tools from arithmetic geometry and holomorphic dynamics, they prove an 'unlikely intersection' statement for such pairs, thereby demonstrating strong rigidity features for them. They further describe one-dimensional families in the moduli space of polynomials containing infinitely many postcritically finite parameters, proving the dynamical André-Oort conjecture for curves in this context, originally stated by Baker and DeMarco.

MATHEMATICS / Geometry / Algebraic. --- Affine plane. --- Affine space. --- Affine transformation. --- Algebraic closure. --- Algebraic curve. --- Algebraic equation. --- Algebraic extension. --- Algebraic surface. --- Algebraic variety. --- Algebraically closed field. --- Analysis. --- Analytic function. --- Analytic geometry. --- Approximation. --- Arithmetic dynamics. --- Asymmetric graph. --- Ball (mathematics). --- Bifurcation theory. --- Boundary (topology). --- Cantor set. --- Characterization (mathematics). --- Chebyshev polynomials. --- Coefficient. --- Combinatorics. --- Complex manifold. --- Complex number. --- Computation. --- Computer programming. --- Conjugacy class. --- Connected component (graph theory). --- Continuous function (set theory). --- Coprime integers. --- Correspondence theorem (group theory). --- Counting. --- Critical graph. --- Cubic function. --- Datasheet. --- Disk (mathematics). --- Divisor (algebraic geometry). --- Elliptic curve. --- Equation. --- Equidistribution theorem. --- Equivalence relation. --- Euclidean topology. --- Existential quantification. --- Fixed point (mathematics). --- Function space. --- Geometric (company). --- Graph (discrete mathematics). --- Hamiltonian mechanics. --- Hausdorff dimension. --- Hausdorff measure. --- Holomorphic function. --- Inequality (mathematics). --- Instance (computer science). --- Integer. --- Intermediate value theorem. --- Intersection (set theory). --- Inverse-square law. --- Irreducible component. --- Iteration. --- Jordan curve theorem. --- Julia set. --- Limit of a sequence. --- Line (geometry). --- Metric space. --- Moduli space. --- Moment (mathematics). --- Montel's theorem. --- P-adic number. --- Parameter. --- Pascal's Wager. --- Periodic point. --- Polynomial. --- Power series. --- Primitive polynomial (field theory). --- Projective line. --- Quotient ring. --- Rational number. --- Realizability. --- Renormalization. --- Riemann surface. --- Ring of integers. --- Scientific notation. --- Set (mathematics). --- Sheaf (mathematics). --- Sign (mathematics). --- Stone–Weierstrass theorem. --- Subharmonic function. --- Support (mathematics). --- Surjective function. --- Theorem. --- Theory. --- Topology. --- Transfer principle. --- Union (set theory). --- Unit disk. --- Variable (computer science). --- Variable (mathematics). --- Zariski topology. --- Polynomials. --- Dynamics. --- Geometry, Algebraic. --- Algebraic geometry --- Geometry --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Algebra --- Algebraic geometry. --- Mathematics.

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An elliptic curve is a particular kind of cubic equation in two variables whose projective solutions form a group. Modular forms are analytic functions in the upper half plane with certain transformation laws and growth properties. The two subjects--elliptic curves and modular forms--come together in Eichler-Shimura theory, which constructs elliptic curves out of modular forms of a special kind. The converse, that all rational elliptic curves arise this way, is called the Taniyama-Weil Conjecture and is known to imply Fermat's Last Theorem. Elliptic curves and the modeular forms in the Eichler- Shimura theory both have associated L functions, and it is a consequence of the theory that the two kinds of L functions match. The theory covered by Anthony Knapp in this book is, therefore, a window into a broad expanse of mathematics--including class field theory, arithmetic algebraic geometry, and group representations--in which the concidence of L functions relates analysis and algebra in the most fundamental ways. Developing, with many examples, the elementary theory of elliptic curves, the book goes on to the subject of modular forms and the first connections with elliptic curves. The last two chapters concern Eichler-Shimura theory, which establishes a much deeper relationship between the two subjects. No other book in print treats the basic theory of elliptic curves with only undergraduate mathematics, and no other explains Eichler-Shimura theory in such an accessible manner.

Curves, Elliptic. --- Affine plane (incidence geometry). --- Affine space. --- Affine variety. --- Algebra homomorphism. --- Algebraic extension. --- Algebraic geometry. --- Algebraic integer. --- Algebraic number theory. --- Algebraic number. --- Analytic continuation. --- Analytic function. --- Associative algebra. --- Automorphism. --- Big O notation. --- Binary quadratic form. --- Birch and Swinnerton-Dyer conjecture. --- Bounded set (topological vector space). --- Change of variables. --- Characteristic polynomial. --- Coefficient. --- Compactification (mathematics). --- Complex conjugate. --- Complex manifold. --- Complex number. --- Conjecture. --- Coprime integers. --- Cusp form. --- Cyclic group. --- Degeneracy (mathematics). --- Dimension (vector space). --- Dirichlet character. --- Dirichlet series. --- Division algebra. --- Divisor. --- Eigenform. --- Eigenvalues and eigenvectors. --- Elementary symmetric polynomial. --- Elliptic curve. --- Elliptic function. --- Elliptic integral. --- Equation. --- Euler product. --- Finitely generated abelian group. --- Fourier analysis. --- Function (mathematics). --- Functional equation. --- General linear group. --- Group homomorphism. --- Group isomorphism. --- Hecke operator. --- Holomorphic function. --- Homomorphism. --- Ideal (ring theory). --- Integer matrix. --- Integer. --- Integral domain. --- Intersection (set theory). --- Inverse function theorem. --- Invertible matrix. --- Irreducible polynomial. --- Isogeny. --- J-invariant. --- Linear fractional transformation. --- Linear map. --- Liouville's theorem (complex analysis). --- Mathematical induction. --- Meromorphic function. --- Minimal polynomial (field theory). --- Modular form. --- Monic polynomial. --- Möbius transformation. --- Number theory. --- P-adic number. --- Polynomial ring. --- Power series. --- Prime factor. --- Prime number theorem. --- Prime number. --- Principal axis theorem. --- Principal ideal domain. --- Principal ideal. --- Projective line. --- Projective variety. --- Quadratic equation. --- Quadratic function. --- Quadratic reciprocity. --- Riemann surface. --- Riemann zeta function. --- Simultaneous equations. --- Special case. --- Summation. --- Taylor series. --- Theorem. --- Torsion subgroup. --- Transcendence degree. --- Uniformization theorem. --- Unique factorization domain. --- Variable (mathematics). --- Weierstrass's elliptic functions. --- Weil conjecture.