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

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. Front Matter....Pages i-xiii An Introduction to Zeta Functions....Pages 1-63 Introduction to Compact Riemann Surfaces, Jacobians, and Abelian Varieties....Pages 64-211 Elliptic Curves....Pages 212-237 Introduction to Modular Forms....Pages 238-291 Decorated Elliptic Curves: Modular Aspects....Pages 292-312 Galois Theory, Algebraic Number Theory, and Zeta Functions....Pages 313-393 Galois Theory for Coverings and Riemann Surfaces....Pages 394-412 Differential Galois Theory....Pages 413-439 p-adic Numbers and Ultrametricity....Pages 440-475 Introduction to Lattice Geometry....Pages 476-495 A Short Introduction to Quasicrystallography....Pages 496-537 Gap Labelling Theorems for Schrödinger Operators....Pages 538-630 Circle Maps: Irrationally Winding....Pages 631-658 An Introduction To Small Divisors Problems....Pages 659-679 Back Matter....Pages 680-690

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.