Local Langlands in Families Morphism

• Local Langlands in Families Morphism is a construction that attaches to a continuous family of Galois representations an admissible smooth GLₙ(E)-module, ensuring torsion-freeness, specialization, and generic cosocle properties.
• Its methodology employs the Breuil–Schneider modification to maintain compatibility with base change, cyclic generation, and an endomorphism ring equal to the coefficient ring.
• The approach underpins robust arithmetic applications by linking deformation theory, modularity lifting, and global tensor product compatibility for patching local data in completed cohomology.

The local Langlands in families morphism is a canonical, functorial assignment that interpolates the classical local Langlands correspondence for general linear groups (GLₙ) across families of Galois representations parameterized by Noetherian local rings. This construction attaches to a continuous family of Galois representations parameterized by a suitable coefficient ring A a single admissible smooth A[GLₙ(E)]-module in a way compatible with the classical correspondence at the characteristic zero points of Spec A, providing a robust foundation for deformation-theoretic, p-adic, and geometric representation-theoretic applications.

1. Foundational Definitions and Structures

Given a nonarchimedean local field E with residue characteristic $l$, and a reduced complete Noetherian local ring $A$ with finite residue field $k$ of characteristic $p \neq l$, a continuous Galois representation

$\rho: G_E \rightarrow GL_n(A)$

is considered, with $A$ assumed flat over the Witt vectors $W(k)$. The central goal is to construct, for each such $\rho$, an admissible smooth $A[GL_n(E)]$-module $V=V_\rho$ (up to canonical isomorphism) subject to these constraints:

• $A$-torsion free: All associated primes of $V$ are minimal. This ensures no unwanted "torsion phenomena" on the geometric or Galois side.
• Specialization property: For every minimal prime $\mathfrak{a} \subset A$, the specialized fiber $V \otimes_A K(\mathfrak{a})$ is isomorphic, up to smooth duality, to the classical (Breuil–Schneider-modified) local Langlands correspondence representation $T(\rho_\mathfrak{a})$ over $K(\mathfrak{a})$.
• Generic cosocle: For the reduction $V/\mathfrak{m}V$ (with $\mathfrak{m}$ the maximal ideal), the cosocle is absolutely irreducible and generic, and the kernel of $$V/\mathfrak{m}V \to \mathrm{cosoc}(V/\mathfrak{m}V)$$ contains no generic constituents.

These properties ensure $V$ behaves well under arithmetic and geometric operations (base-change, specialization, reduction), and that it interpolates the local Langlands correspondence on points of Spec $A$, securing broad functorial compatibility.

2. Existence, Uniqueness, and Structural Theorems

Theorem 6.2.1 establishes that under the hypotheses above, such a $V$ exists (when $A$ is the ring of integers in a finite extension of $\mathbb{Q}_p$) and is unique up to isomorphism, with the following strong additional features:

• $V$ is cyclic as an $A[GL_n(E)]$-module (i.e., generated by a single vector).
• The endomorphism ring of $V$ is exactly $A$.
• $V$ is determined uniquely by its family of specializations and its generic cosocle over the residue field.

Furthermore, the construction is local–global compatible: for a finite set $S$ of nonarchimedean places $v$ (not over $p$) and tuples of Galois representations $\{\rho_v\}_{v \in S}$, the corresponding module for $G = \prod_{v \in S} GL_n(E_v)$ is isomorphic to the normalized tensor product over $v$ of the local modules $V_{\rho_v}$, up to a maximal torsion-free quotient. This coherence across places is crucial for arithmetic and automorphic applications.

Specialization compatibility (Theorems 6.2.5, 6.2.6, Definition 6.2.9): For every prime ideal $\mathfrak{p}\subset A$, there is a natural equivariant map from the Breuil–Schneider representation over $K(\mathfrak{p})$ to the reduction $V\otimes_A K(\mathfrak{p})$, which is an isomorphism in the minimal case.

3. Modified and Interpolative Nature of the Correspondence

The morphism utilizes the "Breuil–Schneider" variant of the local Langlands correspondence, associating (possibly reducible but essentially AIG) representations to Frobenius-semisimple Weil–Deligne representations. This choice, in contrast with classical approaches, has two critical advantages:

• Change of coefficients: The assignment commutes with arbitrary base-change in A.
• Specialization in families: The admissible module $V$ ensures that for any specialization (corresponding to a point $a \in \mathrm{Spec}\,A$), the fiber is the expected local Langlands dual representation.

This defines a true family-theoretic morphism: $\rho \ \mapsto\ \mathfrak{V}_A(\rho)$ which is minimal, cyclic, endomorphism-algebraic, and "interpolatory" in the sense that every other module interpolating the same specializations admits a $GL_n(E)$-equivariant injection from this $V$.

4. Applications and Implications in Arithmetic, Deformation, and Cohomology

The machinery is crucial in settings where families of Galois representations arise. For instance:

• Patching Arguments: In the context of completed cohomology and eigenvarieties, the continuous variation (even rigid-analytic variation across $A$) of the correspondence allows one to "patch together" local data for different $p$-adic families.
• Descent, Uniqueness, and Arithmetic Detection: Uniqueness (minimality, cyclicity, endomorphism ring) ensures that deformation-theoretic and arithmetic properties lift "nicely" from characteristic zero points to the entire family, which is essential for arguments in modularity lifting and the construction of Galois parameter spaces.
• Top Derivative Generation: If $V/\mathfrak{m}V$ is essentially AIG, then $V$ is generated by its "top derivative" (as in the Schwartz space/Kirillov model), a property leveraged for uniqueness and base-change compatibility, as well as in the paper of global completed cohomologies.

5. Relation to Previous and Parallel Works

This construction is a vast generalization of the mod $p$ local Langlands correspondence (e.g., Vignéras for $GL_n$ over $p$-adic fields), the Bernstein–Zelevinski theory of induced representations, and the Breuil–Schneider modification. The "in families" morphism synthesizes these approaches, allowing for coefficient rings that reflect families of Galois representations (e.g. deformation rings), and ensures that all operations (specialization, twist, tensor product, change of coefficients) behave compatibly across the family. Unlike the classical viewpoint, which handles only pointwise Galois representations, this approach allows for the robust global interpolation necessary for $p$-adic automorphic representation theory and congruence applications.

6. Technical Implementation and Limitations

The functoriality and universality of the local Langlands in families morphism depend heavily on the structural properties of $A$, primarily its flatness and reducedness over $W(k)$. For $A$ not the ring of integers of a finite extension of $\mathbb{Q}_p$, existence requires further input about specialization and cosocle generation. Furthermore, for full compatibility in global applications, one must ensure that the local factors glue appropriately (and uniquely) when varying more general coefficient rings and when considering ramification and wild inertia phenomena.

The approach is optimized for settings where $p\neq l$ and $A$ is reduced. While the authors implement the necessary checks for uniqueness and minimality, explicit functorial constructions for all $A$ remain constrained by these structural hypotheses.

7. Summary Table: Key Properties

Property	Description	Mathematical Formalism
Torsion-free over $A$	No non-trivial $A$-torsion in $V$	$A$-module, all associated primes minimal
Specialization Compatibility	Fibers recover Breuil–Schneider correspondence	$$V \otimes_A K(\mathfrak{a}) \simeq T(\rho_{\mathfrak{a}})$$
Minimality and Cyclicity	Any interpolating family injects into $V$; $V$ generated by one element	$V$ cyclic $A[G]$-module; $\mathrm{End}_G(V)=A$
Generic Cosocle	Maximal semisimple quotient (cosocle) over residue field is generic/AIG	$\mathrm{cosoc}(V/\mathfrak{m}V)$ absolutely irreducible
Tensor Product Compatibility	Local modules tensor to form global module (up to torsion-free quotient)	$V \cong \otimes_{v \in S}^\prime V_{\rho_v}$
Change of Coefficients/Base Change	Construction commutes with scalar extension and reduction	$V_{A'} = V_A \otimes_A A'$

This local Langlands in families morphism thus forms a backbone for the modern arithmetic linear representation theory of $p$-adic groups in families, securing the necessary interpolation, specialization, and arithmetic control essential for applications in the cohomology of Shimura varieties, modularity lifting, and the emerging categorical and geometric Langlands frameworks.