Computational Aspects of Modular Forms and Galois Representations: How One Can Compute in Polynomial Time the Value of Ramanujan's Tau at a Prime (AM-176)

Computational Aspects of Modular Forms and Galois Representations: How One Can Compute in Polynomial Time the Value of Ramanujan's Tau at a Prime (AM-176)


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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.

Product Details

ISBN-13: 9780691142029
Publisher: Princeton University Press
Publication date: 06/20/2011
Series: Annals of Mathematics Studies , #176
Pages: 440
Product dimensions: 6.10(w) x 9.20(h) x 1.00(d)

About the Author

Bas Edixhoven is professor of mathematics at the University of Leiden. Jean-Marc Couveignes is professor of mathematics at the University of Toulouse le Mirail. Robin de Jong is assistant professor at the University of Leiden. Franz Merkl is professor of applied mathematics at the University of Munich. Johan Bosman is a postdoctoral researcher at the Institut für Experimentelle Mathematik in Essen, Germany.

Table of Contents

Preface ix

Acknowledgments x

Author information xi

Dependencies between the chapters xii

Chapter 1 Introduction, main results, context B. Edixhoven 1

1.1 Statement of the main results 1

1.2 Historical context: Schoof's algorithm 7

1.3 Schoof's algorithm described in terms of étale cohomology 9

1.4 Some natural new directions 12

1.5 More historical context: congruences for Ramanujan's τ-function 16

1.6 Comparison with p-adic methods 26

Chapter 2 Modular curves, modular forms, lattices, Galois representations B. Edixhoven 29

2.1 Modular curves 29

2.2 Modular forms 34

2.3 Lattices and modular forms 42

2.4 Galois representations attached to eigenforms 46

2.5 Galois representations over finite fields, and reduction to torsion in Jacobians 55

Chapter 3 First description of the algorithms J.-M. Couveignes B. Edixhoven 69

Chapter 4 Short introduction to heights and Arakelov theory B. Edixhoven R. de Jong 79

4.1 Heights on Q and Q 79

4.2 Heights on projective spaces and on varieties 81

4.3 The Arakelov perspective on height functions 86

4.4 Arithmetic surfaces, intersection theory, and arithmetic Riemann-Roch 88

Chapter 5 Computing complex zeros of polynomials and power series J.-M. Couveignes 95

5.1 Polynomial time complexity classes 96

5.2 Computing the square root of a positive real number 101

5.3 Computing the complex roots of a polynomial 107

5.4 Computing the zeros of a power series 115

Chapter 6 Computations with modular forms and Galois representations J. Bosman 129

6.1 Modular symbols 129

6.2 termezzo: Atkin-Lehner operators 138

6.3 asic numerical evaluations 140

6.4 Numerical calculations and Galois representations 150

Chapter 7 Polynomials for projective representations of level one forms J. Bosman 159

7.1 Introduction 159

7.2 Galois representations 161

7.3 Proof of the theorem 166

7.4 Proof of the corollary 167

7.5 The table of polynomials 170

Chapter 8 Description of X1 (5l) B. Edixhoven 173

8.1 Construction of a suitable cuspidal divisor on X1 (5l) 173

8.2 The exact setup for the level one case 178

Chapter 9 Applying Arakelov theory B. Edixhoven R. de Jong 187

9.1 Relating heights to intersection numbers 187

9.2 Controlling Dx-D0 195

Chapter 10 An upper bound for Green functions on Riemann surfaces F. Merkl 2

Chapter 11 Bo0ds for Arakelov invariants of modular curves S.Edixhoven R. de Jong 217

11.1 Bounding the height of X1 (pl) 217

11.2 Bounding the theta function on Picg-1 (X1(pl)) 225

11.3 Upper bounds for Arakelov Green functions on the curves X1 (pl) 232

11.4 Bounds for intersection numbers on X1 (pl) 241

11.5 A bound for h(xl'(Q)) in terms of h (bl (Q)) 244

11.6 An integral over X1 (5l) 246

13.7 Final estimates of the Arakelov contribution 249

Chapter 12 Approximating Vf over the complex numbers J.-M. Couveignes 257

12.1 Points, divisors, and coordinates on X 260

12.2 The lattice of periods 263

12.3 Modular functions 266

12.4 Power series 279

12.5 Jacobian and Wronskian determinants of series 286

12.6 A simple quantitative study of the Jacobi map 292

12.7 Equivalence of various norms 297

12.8 An elementary operation in the Jacobian variety 303

12.9 Arithmetic operations in the Jacobian variety 306

12.10 The inverse Jacobi problem 307

12.11 The algebraic conditioning 313

12.12 Height 319

12.13 Bounding the error in Xg 323

12.14 Final result of this chapter 334

Chapter 13 Computing Vf modulo p J.-M. Couveignes 337

13.1 Basic algorithms for plane curves 338

13.2 A first approach to picking random divisors 346

13.3 Pairings 350

13.4 Divisible groups 354

13.5 The Kummer map 359

13.6 Linearization of torsion classes 362

13.7 Computing Vf modulo p 366

Chapter 14 Computing the residual Galois representations B. Edixhoven 371

14.1 Main result 371

14.2 Reduction to irreducible representations 372

14.3 Reduction to torsion in Jacobians 373

14.4 Computmg the Q(ζl)-algebra corresponding to V 374

14.5 Computing the vector space structure 378

14.6 Descent to Q 379

14.7 Extracting the Galois representation 379

14.8 A probabilistic variant 380

Chapter 15 Computing coefficients of modular forms B. Edixhoven 383

15.1 Computing τ(p) in time polynomial in log p 383

15.2 Computings Tn for large n and large weight 385

15.3 An application to quadratic forms 397

Epilogue 399

Bibliography 403

Index 423

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