brauer groups, tamagawa measures, and rational points

CONTENTS
v
3. On the geometry of diagonal cubic threefolds . . . . . . . . . . . . . . . . . . . . . . 252
4. Accumulating subvarieties . . ........................................ 254
5. Results . . ............................................................ 259
VIII. On the smallest point on a diagonal quartic threefold . . . . . . . . . . 265
1. A computer experiment . . .......................................... 265
2. A negative result . . .................................................. 268
3. The fundamental finiteness property . . .............................. 270
IX. General cubic surfaces . . ............................................ 281
1. Rational points on cubic surfaces . . .................................. 281
2. Background . . ........................................................ 284
3. The Galois operation on the 27 lines . . .............................. 287
4. Computation of Peyre’s constant . . .................................. 290
5. Numerical Data . . .................................................... 292
6. A concrete example . . ................................................ 297
X. On the smallest point on a diagonal cubic surface . . ................ 301
1. Introduction . . ...................................................... 301
2. The factors α and β . . ....................... . ...................... 305
3. Splitting the Picard group . . .......................................... 306
4. A technical lemma . . ................................................ 310
5. A negative result . . .................................................. 311
6. The fundamental finiteness property . . .............................. 316
XI. The Diophantine Equation x 4 + 2y 4 = z 4 + 4w 4 . . . . . . . . . . . . . . . . . . 329
1. Introduction . . ...................................................... 329
2. Congruences . . ...................................................... 330
3. Naive methods . . .................................................... 332
4. An algorithm to efficiently search for solutions . . .................... 332
5. General formulation of the method . . ................................ 335
6. Improvements I – More congruences . . .............................. 336
7. Improvements II – Adaption to our hardware . . . . . . . . . . . . . . . . . . . . . . 341
8. The solution found . . ................................................ 348
XII. New sums of three cubes . . ........................................ 349
1. Introduction . . ...................................................... 349
2. Elkies’ method . . .......................... .......................... 350
3. Implementation . . .................................................... 350