Notes from the International Autumn School on Computational Number Theory

Preface......Page 7
Acknowledgements......Page 8
Contents......Page 9
Lecture Notes......Page 10
 1 Functional Equations......Page 11
  1.1 Fourier Series......Page 12
 2 Elliptic Functions......Page 14
  3.1 Definitions......Page 20
  3.2 Basic Results......Page 22
  3.3 The Scalar Product......Page 25
  3.4 Fourier Expansions......Page 26
  3.5 Obtaining Modular Forms by Averaging......Page 28
  3.6 The Ramanujan Delta Function......Page 30
  3.7 Product Expansions and the Dedekind Eta Function......Page 33
  3.8 Computational Aspects of the Ramanujan τ Function......Page 36
  3.9 Modular Functions and Complex Multiplication......Page 37
  3.10 Derivatives of Modular Forms......Page 40
 4 Hecke Operators: Ramanujan\'s Discoveries......Page 42
  5.1 Euler Products......Page 44
  5.2 Analytic Properties of L-Functions......Page 46
  5.3 Special Values of L-Functions......Page 49
  5.4 Nonanalytic Eisenstein Series and Rankin–Selberg......Page 50
  6.1 Types of Subgroups......Page 53
  6.2 Modular Forms on Subgroups......Page 54
  6.3 Examples of Modular Forms on Subgroups......Page 56
  6.4 Hecke Operators and L-Functions......Page 57
  6.5 Modular Forms with Characters......Page 60
  6.7 Origins of Modular Forms......Page 62
  7.1 Modular Forms of Half-Integral Weight......Page 64
  7.2 Modular Forms in Several Variables......Page 67
 9 Suggestions for Further Reading......Page 69
 References......Page 70
  1.1 Theory: Brief Review of Modular Forms and Hecke Operators......Page 71
  1.2 Theory: The Modular Symbols Formalism......Page 77
  1.3 Theory: The Modular Symbols Algorithm......Page 84
  1.4 Theory: Number Theoretic Applications......Page 87
  1.5 Theory: Exercises......Page 90
  1.6 Computer Exercises......Page 91
 2 Hecke Algebras......Page 93
  2.1 Theory: Hecke Algebras and Modular Forms over Rings......Page 94
  2.2 Algorithms and Implementations: Localisation Algorithms......Page 104
  2.3 Theoretical Exercises......Page 108
  2.4 Computer Exercises......Page 109
  3.1 Theory: Categories and Functors......Page 110
  3.2 Theory: Complexes and Cohomology......Page 113
  3.3 Theory: Cohomological Techniques......Page 116
  3.4 Theory: Generalities on Group Cohomology......Page 121
  3.5 Theoretical Exercises......Page 124
  4.1 Theory: The Standard Fundamental Domain for PSL2(mathbbZ)......Page 126
  4.2 Theory: PSL2(mathbbZ) as a Free Product......Page 128
  4.3 Theory: Mayer–Vietoris for PSL2(mathbbZ)......Page 130
  4.4 Theory: Parabolic Group Cohomology......Page 134
  4.5 Theory: Dimension Computations......Page 135
  4.6 Theoretical Exercises......Page 138
  4.7 Computer Exercises......Page 139
  5.1 Theory: Manin Symbols......Page 140
  5.2 Theory: Manin Symbols and Group Cohomology......Page 144
  5.3 Algorithms and Implementations: Conversion Between Manin and Modular Symbols......Page 145
  5.4 Theoretical Exercises......Page 146
  6.1 Theory: Petersson Scalar Product......Page 147
  6.2 Theory: The Eichler–Shimura Map......Page 152
  6.3 Theory: Cup Product and Petersson Scalar Product......Page 157
  6.4 Theory: The Eichler–Shimura Theorem......Page 163
  7.1 Theory: Hecke Rings......Page 164
  7.2 Theory: Hecke Operators on Modular Forms......Page 168
  7.3 Theory: Hecke Operators on Group Cohomology......Page 170
  7.4 Theory: Hecke Operators and Shapiro\'s Lemma......Page 172
  7.5 Theory: Eichler–Shimura Revisited......Page 173
  7.6 Theoretical Exercises......Page 176
 References......Page 177
  1.1 Introduction......Page 179
  1.2 The Prototype: The Riemann Zeta Function ζ(s)......Page 180
  1.3 Dedekind Zeta Functions......Page 181
  1.5 Examples in Weight 1......Page 184
  1.6 Definition of a Global L-Function......Page 185
  2.1 L-Functions Coming from Modular Forms......Page 187
  2.2 Local L-Functions of Algebraic Varieties......Page 188
  2.3 Global L-Function Attached to a Variety......Page 190
  2.4 Hypergeometric Motives......Page 191
  2.6 Results and Conjectures on L(V;s)......Page 194
  2.7 An Explicit Numerical Example of BSD......Page 196
  2.8 An Explicit Numerical Example of Beilinson–Bloch......Page 198
  2.10 Computational Goals......Page 199
  2.11 Available Software for L-Functions......Page 200
 3 Arithmetic Methods: Computing a(n)......Page 201
  3.2 Elliptic Curves with Complex Multiplication......Page 202
  3.3 Using Modular Forms of Weight 2......Page 204
  3.4 Higher Weight Modular Forms......Page 205
  3.5 Computing |V(mathbbFq)| for Quasi-diagonal Hypersurfaces......Page 206
  4.1 Gauss Sums over mathbbFq......Page 207
  4.2 Jacobi Sums over mathbbFq......Page 209
  4.3 Applications of J(χ,χ)......Page 212
  4.4 The Hasse–Davenport Relations......Page 213
 5 Practical Computations of Gauss and Jacobi Sums......Page 214
  5.1 Elementary Methods......Page 215
  5.2 Sample Implementations......Page 218
  5.3 Using Theta Functions......Page 219
  5.4 Using the Gross–Koblitz Formula......Page 220
  6.1 Definitions......Page 227
  6.2 Reduction to Prime Gauss Sums......Page 229
  6.3 General Complete Exponential Sums over mathbbZ/NmathbbZ......Page 231
  7.1 Computational Issues......Page 232
  7.2 Dirichlet L-Functions......Page 233
  7.3 Approximate Functional Equations......Page 235
  7.4 Inverse Mellin Transforms......Page 239
  7.5 Hadamard Products and Explicit Formulas......Page 242
 8 Some Useful Analytic Computational Tools......Page 247
  8.1 The Euler–Maclaurin Summation Formula......Page 248
  8.3 Zagier\'s Extrapolation Method......Page 250
  8.4 Computation of Euler Sums and Euler Products......Page 252
  8.5 Summation of Alternating Series......Page 253
  8.6 Numerical Differentiation......Page 255
  8.7 Double Exponential Numerical Integration......Page 257
  8.8 The Use of Abel–Plana for Definite Summation......Page 260
  9.1 Introduction......Page 261
  9.2 The Two Basic Algorithms......Page 262
  9.3 Using Continued Fractions for Inverse Mellin Transforms......Page 263
  9.4 Using Continued Fractions for Gaussian Integration and Summation......Page 264
 10 Pari/GP Commands......Page 266
  11.2 The Beilinson–Bloch Example......Page 270
  12.1 The Gamma Function......Page 271
  12.2 Order of a Function: Hadamard Factorization......Page 272
  12.3 Elliptic Curves......Page 273
 References......Page 274
 1 Introduction......Page 275
 2 Binary Recurrent Sequences......Page 276
  3.1 The Fibonacci and Lucas Sequences......Page 278
  3.3 Binary Recurrences Associated to Pell Equations  with N=pm1......Page 279
 4 Cyclotomic Polynomials......Page 281
 5 Lucas Sequences......Page 282
 6 Zsigmondy\'s Theorem......Page 286
 7 The Primitive Divisor Theorem......Page 288
 8 Applications......Page 289
 9 Problems......Page 291
 10 Notes......Page 293
  11.1 Statements......Page 294
  11.3 The p-Adic Case......Page 295
  11.4 Linear Forms in Two Logarithms......Page 296
  11.5 Reducing the Bounds......Page 297
 12 Applications......Page 298
  12.1 Matt\'s Equation......Page 299
  12.2 Rep-Digit Fibonacci Numbers......Page 302
 13 Problems......Page 309
 15 An Application of the LLL Algorithm......Page 312
 References......Page 316
Research Contributions......Page 318
 1 Introduction......Page 319
  2.1 Height Functions......Page 321
  2.2 Ultraproducts, Nonstandard Extensions and Height Functions......Page 323
  2.3 Proper Ideals and Degree Bounds......Page 325
 3 Proof of the Main Remark......Page 326
 References......Page 327
 1 Introduction......Page 329
 2 Preliminaries......Page 330
 3  λ-Constacyclic Codes of Odd Length......Page 331
 4 A New Gray Map and Gray Images of  λ-Constacyclic and Cyclic Codes......Page 332
  4.1 Gray Images of  λ-Constacyclic Codes over the Finite Ring  R......Page 333
 References......Page 334
 1 Introduction......Page 336
 2 Standard Basis of Eisenstein Eigenforms......Page 339
 3 Upper Bound of Congruences......Page 344
 4 Rational Congruences......Page 346
 5 Algorithmic Search for Congruences......Page 348
  5.1 Algorithm......Page 349
 6 Numerical Data......Page 350
  6.1 Description of Data in the Tables......Page 351
 7 Summary of Computational Results......Page 353
 References......Page 357
 1 Introduction......Page 359
 2 Auxiliary Results......Page 361
 3 Main Theorem......Page 362
 References......Page 366