From Number Theory to Physics

1. Front Matter

   Pages i-xiii

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2. An Introduction to Zeta Functions

   • Pierre Cartier

   Pages 1-63

3. Introduction to Compact Riemann Surfaces, Jacobians, and Abelian Varieties

   • Jean-Benoît Bost

   Pages 64-211

4. Elliptic Curves

   • Henri Cohen

   Pages 212-237

5. Introduction to Modular Forms

   • Don Zagier

   Pages 238-291

6. Decorated Elliptic Curves: Modular Aspects

   • Robert Gergondey

   Pages 292-312

7. Galois Theory, Algebraic Number Theory, and Zeta Functions

   • Harold M. Stark

   Pages 313-393

8. Galois Theory for Coverings and Riemann Surfaces

   • Eric Reyssat

   Pages 394-412

9. Differential Galois Theory

   • Frits Beukers

   Pages 413-439

10. p-adic Numbers and Ultrametricity

    • Gilles Christol

    Pages 440-475

11. Introduction to Lattice Geometry

    • Marjorie Senechal

    Pages 476-495

12. A Short Introduction to Quasicrystallography

    • André Katz

    Pages 496-537

13. Gap Labelling Theorems for Schrödinger Operators

    • Jean Bellissard

    Pages 538-630

14. Circle Maps: Irrationally Winding

    • Predrag Cvitanović

    Pages 631-658

15. An Introduction To Small Divisors Problems

    • Jean-Christophe Yoccoz

    Pages 659-679

16. Back Matter

    Pages 680-690

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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.

• Dynamical Systems
• Elliptic Curves
• Galois Theory
• Lattices
• Modular Forms
• Riemann Surfaces
• number theory

• Département de Mathématiques, Université Pierre et Marie Curie (Paris VI), Paris Cedex 05, France

  Michel Waldschmidt

• Centre d’Etudes de Saclay, Service de Physique Théorique, Gif-sur-Yvette Cedex, France

  Pierre Moussa, Jean-Marc Luck, Claude Itzykson

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