vi contents

6. The “first case” of diagonal cubic surfaces 146

7. Concluding remark 161

Part C. Numerical experiments 163

Chapter V. The Diophantine equation

x4

+

2y4

=

z4

+

4w4

165

Numerical experiments and the Manin conjecture 165

1. Introduction 166

2. Congruences 167

3. Naive methods 169

4. An algorithm to eﬃciently search for solutions 169

5. General formulation of the method 171

6. Improvements I—more congruences 172

7. Improvements II—adaptation to our hardware 176

8. The solution found 182

Chapter VI. Points of bounded height on cubic and quartic threefolds 185

1. Introduction—Manin’s conjecture 185

2. Computing the Tamagawa number 189

3. On the geometry of diagonal cubic threefolds 193

4. Accumulating subvarieties 195

5. Results 199

Chapter VII. On the smallest point on a diagonal cubic surface 205

1. Introduction 205

2. Peyre’s constant 208

3. The factors α and β 209

4. A technical lemma 211

5. Splitting the Picard group 212

6. The computation of the L-function at 1 216

7. Computing the Tamagawa numbers 219

8. Searching for the smallest solution 221

9. The fundamental finiteness property 222

10. A negative result 233

Appendix 239

1. A script in GAP 239

2. The list 241

Bibliography 247

Index 261