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This book provides an introduction to the relatively new discipline of arithmetic dynamics. Whereas classical discrete dynamics is the study of iteration of self-maps of the complex plane or real line, arithmetic dynamics is the study of the number-theoretic properties of rational and algebraic points under repeated application of a polynomial or rational function. A principal theme of arithmetic dynamics is that many of the fundamental problems in the theory of Diophantine equations have dynamical analogs. As is typical in any subject combining Diophantine problems and geometry, a fundamental goal is to describe arithmetic properties, at least qualitatively, in terms of underlying geometric structures. Key features: - Provides an entry for graduate students into an active field of research - Provides a standard reference source for researchers - Includes numerous exercises and examples - Contains a description of many known results and conjectures, as well as an extensive glossary, bibliography, and index This graduate-level text assumes familiarity with basic algebraic number theory. Other topics, such as basic algebraic geometry, elliptic curves, nonarchimedean analysis, and the theory of Diophantine approximation, are introduced and referenced as needed. Mathematicians and graduate students will find this text to be an excellent reference.

Mathematics. --- Dynamical Systems and Ergodic Theory. --- Differentiable dynamical systems. --- Mathématiques --- Dynamique différentiable --- Dynamics --- Number theory --- Applied Mathematics --- Calculus --- Engineering & Applied Sciences --- Mathematics --- Physical Sciences & Mathematics --- Number study --- Numbers, Theory of --- Dynamical systems --- Kinetics --- Data structures (Computer science). --- Dynamics. --- Ergodic theory. --- Data Structures. --- Algebra --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Data structures (Computer scienc. --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Data structures (Computer science) --- Information structures (Computer science) --- Structures, Data (Computer science) --- Structures, Information (Computer science) --- Electronic data processing --- File organization (Computer science) --- Abstract data types (Computer science) --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems. --- Artificial intelligence --- Dynamical Systems. --- Data Science. --- Data processing.

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The theory of elliptic curves is distinguished by its long history and by the diversity of the methods that have been used in its study. This book treats the arithmetic theory of elliptic curves in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. The book begins with a brief discussion of the necessary algebro-geometric results, and proceeds with an exposition of the geometry of elliptic curves, the formal group of an elliptic curve, elliptic curves over finite fields, the complex numbers, local fields, and global fields. The last two chapters deal with integral and rational points, including Siegel's theorem and explicit computations for the curve Y 2 = X 3 + DX. The book contains three appendices: Elliptic Curves in Characteristics 2 and 3, Group Cohomology, and a third appendix giving an overview of more advanced topics.

Arithmetic. --- Curves, Algebraic. --- Curves, Elliptic. --- Curves, Elliptic --- Curves, Algebraic --- Courbes algébriques --- 512.74 --- Elliptic curves --- Numbers, Real --- Algebraic groups. Abelian varieties --- 512.74 Algebraic groups. Abelian varieties --- Courbes algébriques --- Arithmetic --- Algebraic curves --- Algebraic varieties --- Mathematics --- Set theory --- Calculators --- Algebraic geometry --- Courbes elliptiques --- Arithmétique --- Arithmetical algebraic geometry --- Géométrie algébrique arithmétique --- Géométrie algébrique --- Géométrie algébrique arithmétique. --- Géométrie algébrique --- Nombres, Théorie des

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A Friendly Introduction to Number Theory, Fourth Edition is designed to introduce students to the overall themes and methodology of mathematics through the detailed study of one particular facet—number theory. Starting with nothing more than basic high school algebra, students are gradually led to the point of actively performing mathematical research while getting a glimpse of current mathematical frontiers. The writing is appropriate for the undergraduate audience and includes many numerical examples, which are analyzed for patterns and used to make conjectures. Emphasis is on the methods used for proving theorems rather than on specific results.

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Algebraic geometry --- Curves, Elliptic --- Diophantine analysis --- Courbes elliptiques --- Analyse diophantienne --- Rational points (Geometry) --- Curves, Elliptic. --- Diophantine analysis. --- Rational points (Geometry). --- 511 --- 511 Number theory --- Number theory --- Indeterminate analysis --- Forms, Quadratic --- Points, Rational (Geometry) --- Arithmetical algebraic geometry --- Elliptic curves --- Curves, Algebraic