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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.
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 X_{1} (5l) B. Edixhoven 173
8.1 Construction of a suitable cuspidal divisor on X_{1} (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 D_{x}-D_{0} 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 X_{1} (pl) 217
11.2 Bounding the theta function on Pic^{g-1} (X_{1}(pl)) 225
11.3 Upper bounds for Arakelov Green functions on the curves X_{1} (pl) 232
11.4 Bounds for intersection numbers on X_{1} (pl) 241
11.5 A bound for h(x_{l}^{'}(Q)) in terms of h (b_{l} (Q)) 244
11.6 An integral over X_{1} (5l) 246
13.7 Final estimates of the Arakelov contribution 249
Chapter 12 Approximating V_{f} 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 X^{g} 323
12.14 Final result of this chapter 334
Chapter 13 Computing V_{f} 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 V_{f} 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(ζ_{l})-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 T_{n} for large n and large weight 385
15.3 An application to quadratic forms 397
Epilogue 399
Bibliography 403
Index 423
Overview
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. ...