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Covers the most recent results around Fermat's Theorem (Andrew Wiles) and the Langlands Conjecture (Lafforgue)
This book surveys from a unified point of view both the modern state and the trends of continuing development of various branches of number theory. Motivated by elementary problems (including some modern areas such as cryptography, factorization and primality testing), the central ideas of modern theories are exposed: algebraic number theory, calculations and properties of Galois groups, non-Abelian generalizations of class field theory, recursive computability and links with Diophantine equations, the arithmetic of algebraic varieties, connections with modular forms, zeta- and L-functions. The authors have tried to present the most significant results and methods of modern time. An overview of the major conjectures is also given in order to illustrate current thinking in number theory. Most of these conjectures are based on analogies between functions and numbers, and on connections with other branches of mathematics such as algebraic topology, analysis, representation theory and geometry
Content Level »Research
Keywords »Arakelov geometry - Arithmetic der algebraischen Zahlen - Elementare Zahlentheorie - Elementary number theory - Langlands program - Langlands-Programm - Modular forms - Non-commutative geometry - arithmetic of algebraic numbers - diophantine equations - diophantische Gleichungen - elliptic curves - elliptische Kurven - logic - public - public key Verschlüsselungssysteme - public key cryptosystems - zeta-functions
I. Problems and Tricks.- 1. Elementary Number Theory.- 2. Some Modern Problems of Elementary Number Theory.- II. Ideas and Theories.- 1. Induction and Recursion.- 2. Arithmetic of Algebraic Numbers.- 3. Arithmetic of Algebraic Varieties.- 4. Zeta Functions and Modular Forms.- References.