Paper XIIStatements and proofs
Statements and proofs
Find a statement anywhere in this work by title, printed number, type or source label. Entries follow the chapter order. Each proof link opens the written proof passage.
Statements by chapter
Global deformation data and exact boundaries
Local conditions are quotient packets
Adequacy and the residual input
Auxiliary group algebras and augmentation
Ordinary and derived patching support
- Definition 6.1: Support and near faithfulness
- Theorem 6.2: Finite module support Read proof
- Theorem 6.3: Fitting support criterion Read proof
- Theorem 6.4: Higher Fitting strata Read proof
- Proposition 6.5: Fitting ideals and base change Read proof
- Remark 6.6
- Theorem 6.7: Near faithfulness is full support Read proof
- Corollary 6.8: Support on every irreducible component Read proof
- Theorem 6.9: Support after a quotient Read proof
- Definition 6.10: Ordinary patching datum
- Theorem 6.11: Ordinary patching support descent Read proof
- Corollary 6.12: Componentwise finite level occupancy Read proof
- Example 6.13: Freeness over auxiliary variables is not enough
- Definition 6.14: Derived support
- Standard fact 6.15: Cohomological description of derived support
- Theorem 6.16: Derived quotient support Read proof
- Definition 6.17: Derived patching datum
- Theorem 6.18: Derived patching support descent Read proof
- Corollary 6.19: A nonzero cohomological degree survives Read proof
- Remark 6.20
Transport through an isomorphism R equals T
- Theorem 7.1: Support and Fitting transport Read proof
- Corollary 7.2: Full support is invariant under (R cong TT ) Read proof
- Proposition 7.3: A quotient can erase a component Read proof
- Theorem 7.4: Field fiber criterion Read proof
- Corollary 7.5: Supported deformation points give nonzero Hecke fibers Read proof
Finite Hecke packets and infinite occupancy
- Standard fact 8.1: Artin decomposition
- Theorem 8.2: Canonical Hecke packet decomposition Read proof
- Corollary 8.3: Simultaneous generalized eigenspaces Read proof
- Corollary 8.4: Rank and trace detection in characteristic zero Read proof
- Remark 8.5
- Example 8.6: A nonsemisimple Hecke packet
- Theorem 8.7: Nested packets in one finite carrier Read proof
- Corollary 8.8: Finite witness of incompatibility Read proof
- Definition 8.9: Stable core
- Lemma 8.10: Surjectivity on stable cores Read proof
- Theorem 8.11: Nonzero stable core criterion Read proof
- Corollary 8.12: Surjective systems Read proof
- Example 8.13: Tail escape in an infinite carrier
Pointwise automorphy from support
- Definition 9.1: Automorphic Hecke datum
- Standard fact 9.2: Chebotarev and Brauer Nesbitt identification
- Theorem 9.3: Support gives a Hecke eigensystem Read proof
- Theorem 9.4: Pointwise Galois realization Read proof
- Corollary 9.5: Full support gives pointwise automorphy on the realized locus Read proof
- Theorem 9.6: Derived pointwise realization Read proof
- Remark 9.7
Compatible systems and companion completion
- Definition 10.1: Algebraic Frobenius spine
- Definition 10.2: Finite companion forest
- Theorem 10.3: Companion branch criterion Read proof
- Corollary 10.4: Empty finite companion obstruction Read proof
- Theorem 10.5: Conditional simultaneous companion completion Read proof
- Standard fact 10.6: Strict compatibility and one member determination
- Theorem 10.7: One member propagation in a fixed system Read proof
- Remark 10.8
- Proposition 10.9: Finite Frobenius separator Read proof
The coefficient prime two
Conditional automorphic completion
- Definition 12.1: Ordinary completion datum at a point
- Theorem 12.2: Ordinary conditional automorphic completion Read proof
- Definition 12.3: Derived completion datum at a point
- Theorem 12.4: Derived conditional automorphic completion Read proof
- Theorem 12.5: Compatible system completion from one automorphic member Read proof
- Theorem 12.6: Five gate formulation Read proof