Author S.A. Stepanov thoroughly investigates the current state of the theory of Diophantine equations and its related methods. Discussions focus on arithmetic, algebraic-geometric, and logical aspects of the problem. Designed for students as well as researchers, the book includes over 250 excercises accompanied by hints, instructions, and references. Written in a clear manner, this text does not require readers to have special knowledge of modern methods of algebraic geometry.

Equations over Finite Fields. Distribution of Quadratic Residues and Nonresidues. Historical Commentaries on Chapters 1 and 2. Rational Points on Algebraic Curves. The Riemann-Roch Theorem. The Riemann Hypothesis for Congruence Zetafunctions. Integral Points on Curves and Nonstandard Arithmetic. The Siegel-Mahler Theorem. Appendix: Hilbert's Tenth Problem. Index.