Patching Support, Hecke Packets, and Conditional Automorphic Completion

Abstract

Let \(\bar\rho\) be a residual Galois representation and let \(R\) be a global deformation ring with declared local conditions. We separate the global automorphy problem into residual occupancy, deformation support, Hecke occupancy, and arithmetic realization. Local conditions are encoded by quotient rings rather than by idempotents, since irreducible components of a local deformation ring need not be clopen. For an ordinary or derived patching datum, full support descends through the augmentation quotient. Near faithfulness is shown to be equivalent to full support for finite modules over Noetherian rings. An isomorphism \(R\cong{\mathbb T}\) transports support, Fitting ideals, and point fibers, but a quotient map can erase an entire component. Finite Hecke algebras are resolved into canonical Artin packets, so simultaneous generalized eigenspaces are available without assuming semisimplicity. Nested packets in one finite carrier stabilize; for changing finite carriers, nonzero stable cores give compatible inverse limit vectors. These results give a pointwise criterion: a characteristic zero deformation point in automorphic support produces a Hecke eigensystem. With attached Galois representations, local global compatibility, and classicality, the point is automorphic. A fixed compatible system then propagates the conclusion to every coefficient place. For finite coefficient quotients, compatible conjugacy classes admit compatible continuous representatives. Declared polynomial quotient conditions survive their adic reconstruction. These completion statements require explicit finite occupancy, continuity, and integral Frobenius data. Universal companion existence and unrestricted reciprocity are not proved.