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Computational Aspects of Modular Forms and Galois Representations

Book Description

Modular forms are tremendously important in various areas of mathematics, from number theory and algebraic geometry to combinatorics and lattices. Their Fourier coefficients, with Ramanujan's tau-function as a typical example, have deep arithmetic significance. Prior to this book, the fastest known algorithms for computing these Fourier coefficients took exponential time, except in some special cases. The case of elliptic curves (Schoof's algorithm) was at the birth of elliptic curve cryptography around 1985. This book gives an algorithm for computing coefficients of modular forms of level one in polynomial time. For example, Ramanujan's tau of a prime number p can be computed in time bounded by a fixed power of the logarithm of p. Such fast computation of Fourier coefficients is itself based on the main result of the book: the computation, in polynomial time, of Galois representations over finite fields attached to modular forms by the Langlands program. Because these Galois representations typically have a nonsolvable image, this result is a major step forward from explicit class field theory, and it could be described as the start of the explicit Langlands program.

The computation of the Galois representations uses their realization, following Shimura and Deligne, in the torsion subgroup of Jacobian varieties of modular curves. The main challenge is then to perform the necessary computations in time polynomial in the dimension of these highly nonlinear algebraic varieties. Exact computations involving systems of polynomial equations in many variables take exponential time. This is avoided by numerical approximations with a precision that suffices to derive exact results from them. Bounds for the required precision--in other words, bounds for the height of the rational numbers that describe the Galois representation to be computed--are obtained from Arakelov theory. Two types of approximations are treated: one using complex uniformization and another one using geometry over finite fields.

The book begins with a concise and concrete introduction that makes its accessible to readers without an extensive background in arithmetic geometry. And the book includes a chapter that describes actual computations.

Table of Contents

  1. Cover
  2. Title
  3. Copyright
  4. Contents
  5. Preface
  6. Acknowledgments
  7. Author information
  8. Dependencies between the chapters
  9. Chapter 1. Introduction, main results, context by B. Edixhoven
    1. 1.1 Statement of the main results
    2. 1.2 Historical context: Schoof’s algorithm
    3. 1.3 Schoof’s algorithm described in terms of étale cohomology
    4. 1.4 Some natural new directions
    5. 1.5 More historical context: congruences for Ramanujan’s τ-function
    6. 1.6 Comparison with p-adic methods
  10. Chapter 2. Modular curves, modular forms, lattices, Galois representations by B. Edixhoven
    1. 2.1 Modular curves
    2. 2.2 Modular forms
    3. 2.3 Lattices and modular forms
    4. 2.4 Galois representations attached to eigenforms
    5. 2.5 Galois representations over finite fields, and reduction to torsion in Jacobians
  11. Chapter 3. First description of the algorithms by J.-M. Couveignes and B. Edixhoven
  12. Chapter 4. Short introduction to heights and Arakelov theory by B. Edixhoven and R. de Jong
    1. 4.1 Heights on Q and Q
    2. 4.2 Heights on projective spaces and on varieties
    3. 4.3 The Arakelov perspective on height functions
    4. 4.4 Arithmetic surfaces, intersection theory, and arithmetic Riemann-Roch
  13. Chapter 5. Computing complex zeros of polynomials and power series by J.-M. Couveignes
    1. 5.1 Polynomial time complexity classes
    2. 5.2 Computing the square root of a positive real number
    3. 5.3 Computing the complex roots of a polynomial
    4. 5.4 Computing the zeros of a power series
  14. Chapter 6. Computations with modular forms and Galois representations by J. Bosman
    1. 6.1 Modular symbols
    2. 6.2 Intermezzo: Atkin-Lehner operators
    3. 6.3 Basic numerical evaluations
    4. 6.4 Numerical calculations and Galois representations
  15. Chapter 7. Polynomials for projective representations of level one forms by J. Bosman
    1. 7.1 Introduction
    2. 7.2 Galois representations
    3. 7.3 Proof of the theorem
    4. 7.4 Proof of the corollary
    5. 7.5 The table of polynomials
  16. Chapter 8. Description of X1(5l) by B. Edixhoven
    1. 8.1 Construction of a suitable cuspidal divisor on X1(5l)
    2. 8.2 The exact setup for the level one case
  17. Chapter 9. Applying Arakelov theory by B. Edixhoven and R. de Jong
    1. 9.1 Relating heights to intersection numbers
    2. 9.2 Controlling Dx – D0
  18. Chapter 10. An upper bound for Green functions on Riemann surfaces by F. Merkl
  19. Chapter 11. Bounds for Arakelov invariants of modular curves by B. Edixhoven and R. de Jong
    1. 11.1 Bounding the height of X1(pl)
    2. 11.2 Bounding the theta function on Picg–1(X1(pl))
    3. 11.3 Upper bounds for Arakelov Green functions on the curves X1(pl)
    4. 11.4 Bounds for intersection numbers on X1(pl)
    5. 11.5 A bound for h(x′l(Q)) in terms of h(bl(Q))
    6. 11.6 An integral over X1(5l)
    7. 11.7 Final estimates of the Arakelov contribution
  20. Chapter 12. Approximating Vf over the complex numbers by J.-M. Couveignes
    1. 12.1 Points, divisors, and coordinates on X
    2. 12.2 The lattice of periods
    3. 12.3 Modular functions
    4. 12.4 Power series
    5. 12.5 Jacobian and Wronskian determinants of series
    6. 12.6 A simple quantitative study of the Jacobi map
    7. 12.7 Equivalence of various norms
    8. 12.8 An elementary operation in the Jacobian variety
    9. 12.9 Arithmetic operations in the Jacobian variety
    10. 12.10 The inverse Jacobi problem
    11. 12.11 The algebraic conditioning
    12. 12.12 Heights
    13. 12.13 Bounding the error in Xg
    14. 12.14 Final result of this chapter
  21. Chapter 13. Computing Vƒ modulo p by J.-M. Couveignes
    1. 13.1 Basic algorithms for plane curves
    2. 13.2 A first approach to picking random divisors
    3. 13.3 Pairings
    4. 13.4 Divisible groups
    5. 13.5 The Kummer map
    6. 13.6 Linearization of torsion classes
    7. 13.7 Computing Vƒ modulo p
  22. Chapter 14. Computing the residual Galois representations by B. Edixhoven
    1. 14.1 Main result
    2. 14.2 Reduction to irreducible representations
    3. 14.3 Reduction to torsion in Jacobians
    4. 14.4 Computing the Q.(ζl)-algebra corresponding to V
    5. 14.5 Computing the vector space structure
    6. 14.6 Descent to Q
    7. 14.7 Extracting the Galois representation
    8. 14.8 A probabilistic variant
  23. Chapter 15. Computing coefficients of modular forms by B. Edixhoven
    1. 15.1 Computing τ(p) in time polynomial in log p
    2. 15.2 Computing Tn for large n and large weight
    3. 15.3 An application to quadratic forms
  24. Epilogue
  25. Bibliography
  26. Index