vi CONTENTS
5.3. 5—invariants for subgroups of PGL2(Zp) 147
Part 3. Applications 159
Chapter 6. Spherical correspondences 161
6.1. Spherical correspondences over R and their cycles 161
6.2. S—sections of bundles on the projective line 165
6.3. Case T trivial: 6—invariants and S—cohomology 167
6.4. Case T non-trivial: £—invariants, £—cohomology and 6—fiber 173
6.5. Case (r, r) solvable 178
6.6. A converse theorem: biquadratic correspondences 181
Chapter 7. Flat correspondences 185
7.1. Flat correspondences over R: 5—line bundles and cycles 185
7.2. 5—characters 190
7.3. 6—invariants 205
7.4. (5-base loci 210
7.5. S—cohomology 212
7.6. The relative generic S—fiber 221
7.7. Converse theorems: quadratic dynamical systems 223
Chapter 8. Hyperbolic correspondences 227
8.1. Review of Abelian schemes and their crystals 228
8.2. Hecke correspondences over R: 5—line bundles and cycles 240
8.3. S—Serre-Tate expansion maps and 6—Serre operators 251
8.4. Constructions of £—invariants 260
8.5. 6—invariants in the ordinary case 275
8.6. S—invariants in the non-ordinary case 279
8.7. 5—cohomology 285
8.8. The relative generic 8—fiber 288
8.9. Hecke correspondences with a regular Hecke n—cycle 292
8.10. A converse theorem: non-rational hyperbolic uniformization 295
List of Results 299
Bibliography 301
Index
307
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