2 Global deformation data and exact boundaries
On this page · Permalink · Source line 196
Throughout, \(S\) is a finite set of places. Frobenius elements are geometric Frobenius elements; all Hecke polynomials, attached representations, and common algebraic spines use that same convention and the normalization specified by the declared realization theorem.
Let \(F\) be a number field. Fix a prime \(p\), a finite extension \(E/{\mathbb Q}_p\), its valuation ring \({\mathcal O}\), uniformizer \(\varpi\), and residue field \(k\). Let
be a continuous residual representation, where \(S\) is a finite set of places containing every place at which the deformation problem is allowed to ramify.
Definition 2.1: Global deformation datum
A global deformation datum consists of the following objects.
A residual representation \(\bar\rho\).
For every \(v\in S\), a framed local deformation ring \(R_v^\square\).
A quotient \(R_v^{\tau_v}\) representing the chosen local condition whenever such a quotient is known to exist.
A global deformation ring \(R^\square\) with a continuous local map
\[ R^{\mathrm{loc}}:={\mathbin{\widehat\otimes}}_{v\in S,{\mathcal O}}R_v^\square \longrightarrow R^\square. \]The induced global ring with local conditions
\[ R^{\tau}:=R^\square{\mathbin{\widehat\otimes}}_{R^{\mathrm{loc}}}R^{\mathrm{loc},\tau}, \qquad R^{\mathrm{loc},\tau}:={\mathbin{\widehat\otimes}}_{v\in S,{\mathcal O}}R_v^{\tau_v}. \]
Permalink · Source line 207 · Claim record · Lean coverage · Code · Worked examples
Representability of deformation functors and the construction of local condition quotients require hypotheses. The standard foundations are due to Mazur and Schlessinger, with many later refinements for local conditions and pseudorepresentations [16], [21], [4].
Standard fact 2.2: Represented deformation problem
Under the usual finiteness and Schur type hypotheses, framed lifts of \(\bar\rho\) are represented by complete Noetherian local \({\mathcal O}\) algebras. A declared local deformation condition is represented by a quotient when the corresponding subfunctor is representable. Tangent and obstruction spaces are described by the appropriate first and second cohomology groups.
Permalink · Source line 229 · Claim record · Lean coverage · Code · Worked examples
Standard fact 2.3: Automorphy lifting package
Published automorphy lifting theorems begin with residual automorphic occupancy and impose hypotheses on the residual image, local deformation conditions, regularity, and the global deformation problem. They then use ordinary or derived patching to compare a deformation ring with a Hecke algebra and to prove automorphy for characteristic zero points in the supported locus. The precise assumptions depend on the theorem being used [8], [3], [6], [25], [17], [1].
Permalink · Source line 233 · Claim record · Lean coverage · Code · Worked examples
The article treats Standard fact 2.2 and Standard fact 2.3 as external inputs. Its results begin after the relevant rings, modules, complexes, and comparison maps have been supplied.
Definition 2.4: Residual automorphic occupancy
The residual datum \((\bar\rho,\tau)\) is automorphically occupied when a nonzero localized automorphic or cohomological Hecke carrier realizes the residual representation and the declared local conditions.
Permalink · Source line 239 · Claim record · Lean coverage · Code · Worked examples
This is a depth zero hypothesis. A positive depth deformation argument above another residual root cannot produce it.
