From number theory to physics [lectures given at the Meeting "Number Theory and Physics", held at the "Centre de Physique", LesHouches, France, March 7 - 16, 1989

Various developments in physics have involved many questions related to number theory, in an increasingly direct way. This trend is especially visible in two broad families of problems, namely, field theories, and dynamical systems and chaos. The 14 chapters of this book are extended, self-contained versions of expository lecture courses given at a school on "Number Theory and Physics" held at Les Houches for mathematicians and physicists. Most go as far as recent developments in the field. Some adapt an original pedagogical viewpoint. The present book contains fourteen expository contributions on various topics connected to Number Theory, or Arithmetics, and its relationships to Theoreti­ cal Physics. The first part is mathematically oriented; it deals mostly with ellip­ tic curves, modular forms, zeta functions, Galois theory, Riemann surfaces, and p-adic analysis. The second part reports on matters with more direct physical interest, such as periodic and quasiperiodic lattices, or classical and quantum dynamical systems. The contribution of each author represents a short self-contained course on a specific subject. With very few prerequisites, the reader is offered a didactic exposition, which follows the author's original viewpoints, and often incorpo­ rates the most recent developments. As we shall explain below, there are strong relationships between the different chapters, even though every single contri­ bution can be read independently of the others. This volume originates in a meeting entitled Number Theory and Physics, which took place at the Centre de Physique, Les Houches (Haute-Savoie, France), on March 7 - 16, 1989. The aim of this interdisciplinary meeting was to gather physicists and mathematicians, and to give to members of both com­ munities the opportunity of exchanging ideas, and to benefit from each other's specific knowledge, in the area of Number Theory, and of its applications to the physical sciences. Physicists have been given, mostly through the program of lectures, an exposition of some of the basic methods and results of Num­ ber Theory which are the most actively used in their branch. Contents......Page all_1263_to_00701.cpc0010.djvu Preface......Page all_1263_to_00701.cpc0002.djvu 1. An Introduction to Zeta Functions......Page all_1263_to_00701.cpc0012.djvu 2. Introduction to Compact Riemann Surfaces, Jacobians, and Abelian Varieties......Page all_1263_to_00701.cpc0075.djvu 3. Elliptic Curves......Page all_1263_to_00701.cpc0223.djvu 4. Introduction to Modular Forms......Page all_1263_to_00701.cpc0249.djvu 5. Decorated Elliptic Curves: Modular Aspects......Page all_1263_to_00701.cpc0303.djvu 6. Galois Theory, Algebraic Number Theory, and Zeta Functions......Page all_1263_to_00701.cpc0324.djvu 7. Galois Theory for Coverings and Riemann Surfaces......Page all_1263_to_00701.cpc0405.djvu 8. Differential Galois Theory......Page all_1263_to_00701.cpc0424.djvu 9. p-adic Numbers and Ultrametricity......Page all_1263_to_00701.cpc0451.djvu 10. Introduction to Lattice Geometry......Page all_1263_to_00701.cpc0487.djvu 11. A Short Introduction to Quasicrystallography......Page all_1263_to_00701.cpc0507.djvu 12. Gap Labelling Theorems for Schrodinger Operators......Page all_1263_to_00701.cpc0549.djvu 13. Circle Maps: Irrationally Winding......Page all_1263_to_00701.cpc0642.djvu 14. An Introduction to Small Divisors Problems......Page all_1263_to_00701.cpc0670.djvu Index......Page all_1263_to_00701.cpc0691.djvu Ch. 1. An Introduction To Zeta Functions / P. Cartier -- Ch. 2. Introduction To Compact Riemann Surfaces, Jacobians, And Abelian Varieties / J.-b. Bost -- Ch. 3. Elliptic Curves / H. Cohen -- Ch. 4. Introduction To Modular Forms / D. Zagier -- Ch. 5. Decorated Elliptic Curves: Modular Aspects / R. Gergondey -- Ch. 6. Galois Theory, Algebraic Number Theory, And Zeta Functions / H. M. Stark -- Ch. 7. Galois Theory For Coverings And Riemann Surfaces / E. Reyssat -- Ch. 8. Differential Galois Theory / F. Beukers -- Ch. 9. P-adic Numbers And Ultrametricity / G. Christol -- Ch. 10. Introduction To Lattice Geometry / M. Senechal -- Ch. 11. A Short Introduction To Quasicrystallography / A. Katz -- Ch. 12. Gap Labelling Theorems For Schrodinger Operators / J. Bellissard -- Ch. 13. Circle Maps: Irrationally Winding / P. Cvitanovic -- Ch. 14. An Introduction To Small Divisors Problems / J.-c. Yoccoz. M. Waldschmidt ... [et Al.] (eds.) ; With Contributions By P. Cartier ... [et Al.]. Lectures Given At The Meeting 'number Theory And Physics' Held At The Centre De Physique, Les Houches, France, March 7-16, 1989--t.p. Verso. Includes Bibliographical References And Index. In this Chapter, we aim at giving an elementary introduction to some functions which were found useful in number theory.