Local-Global Compatibility

• Local-global compatibility is the precise relationship between local Langlands correspondences and global automorphic or Galois representations within the Langlands program.
• Methodologies like Shimura variety cohomology, eigenvarieties, and patching techniques are employed to match local parameters with global automorphic forms.
• This compatibility framework is pivotal for extracting arithmetic invariants, advancing p-adic theories, and addressing challenges in both mod-p and torsion representation contexts.

Local-global compatibility refers to the precise relationship between the behavior of arithmetic or automorphic objects at local places and their global structure, particularly as it arises in the context of the Langlands program for reductive groups over number fields and function fields. Formally, local-global compatibility describes how the local correspondences (such as the local Langlands correspondence connecting smooth representations of a reductive group over local fields to Weil–Deligne or L-parameters) are reflected in the global objects (such as automorphic representations or Galois representations attached to automorphic forms). It asserts that the global parameters constructed from automorphic representations (or related objects) restrict to the expected local parameters at each place, up to carefully specified equivalence (semisimplification, monodromy, etc).

1. Foundational Definitions and Conjectural Frameworks

The local-global compatibility conjecture manifests across several frameworks, universally requiring two ingredients:

• The existence of global parameters: For number fields, global Galois representations attached to automorphic forms (e.g., $$r_\iota(\pi): G_F \to GL_n(\overline{\mathbb{Q}}_l)$$ for a regular algebraic cuspidal automorphic representation $\pi$),
• The local Langlands correspondence: At each local place $v$ (either archimedean or non-archimedean), an explicit correspondence between smooth admissible representations $\pi_v$ of $GL_n(F_v)$ (or more general groups) and $n$-dimensional Weil–Deligne or L-parameters.

Local-global compatibility asserts that, for each place $v$ of $F$, the restriction to decomposition group $G_{F_v}$ of a global Galois parameter $r_\iota(\pi)$ has Weil–Deligne parameter matching the local Langlands correspondence applied to $\pi_v$,

$$\mathrm{WD}(r_\iota(\pi)|_{G_{F_v}})^{F\text{-ss}} \simeq \mathrm{rec}_{F_v}(\pi_v \otimes |\det|_v^{(1-n)/2})$$

with various nuances regarding equivalence (semisimplification, monodromy operator $N$, compatibility with filtration, etc.) depending on context (Matsumoto, 2023, Varma, 2014, Barnet-Lamb et al., 2011, Caraiani, 2012, Yang, 29 Jun 2024).

Compatibility is also formulated for mod-$p$ correspondences and automorphic Galois representations in the torsion setting (Liu et al., 2021, Liu, 2021, Enns et al., 2021).

2. Local-global Compatibility at $\ell = p$ and $\ell \neq p$

The local-global compatibility problem is especially delicate at places dividing the residue characteristic of the coefficient field ($\ell = p$). Several families of results:

• For $\ell \neq p$, local-global compatibility is established up to Frobenius-semismplification for regular algebraic cuspidal automorphic representations, with detailed control of monodromy and purity properties (Varma, 2014, Caraiani, 2010).
• For $\ell = p$, major theorems under Shin-regularity or Iwahori-level hypotheses establish matching of Weil–Deligne parameters up to semisimplification or Frobenius-semismplification (Barnet-Lamb et al., 2011, Barnet-Lamb et al., 2011, Caraiani, 2012, Yang, 29 Jun 2024, Hevesi, 2023).

A more granular result is the control of the monodromy operator $N$. In the Steinberg case (twists of the Steinberg representation), local-global compatibility asserts that global Galois representations exhibit nontrivial monodromy, matching the local Langlands prediction. This is established with automorphy lifting and potential automorphy techniques for rank two and higher (Yang, 29 Jun 2024, Matsumoto, 2023, Yang, 2021, Newton, 2014).

3. Methodologies for Proving Compatibility

A variety of sophisticated global-to-local and patching methods have been developed:

• Shimura variety cohomology: Automorphic forms are realized in the étale or log-crystalline cohomology of Shimura varieties (classical, integral, or perfectoid), and comparison theorems (Rapoport–Zink, Mokrane, Taylor–Yoshida) are leveraged to relate the monodromy actions (from nearby cycles) to Galois representations (Caraiani, 2012, Caraiani, 2010, Barnet-Lamb et al., 2011, Barnet-Lamb et al., 2011).
• Eigenvarieties and interpolation: Rigid analytic families (eigenvarieties) interpolate the local–global compatibility over $p$-adic families, extending beyond classical points (Johansson et al., 2017).
• Patching and automorphy lifting: Taylor–Wiles–Kisin patching, degree-shifting, and potential automorphy techniques enable the construction of congruence ideal Galois representations that satisfy local conditions and match automorphic local types (Matsumoto, 2023, Yang, 2021, Hevesi, 2023, Salazar et al., 13 Aug 2025).
• Completed cohomology and ordinary theory: The spaces of $p$-adic automorphic forms (completed cohomology, ordinary parts) exhibit a rich interaction with local factors, allowing coinvariant descriptions aligned with Hida theory and the completed Kirillov model (Howe, 30 Jun 2025, Chojecki et al., 2014).
• Excursion algebra for function fields: For positive characteristic fields, morphisms of global and local moduli spaces of shtukas (V. Lafforgue, Fargues–Scholze) provide a uniformization enabling compatibility of global parameters and local L-parameters (Li-Huerta, 2023).

4. Compatibility for Mod $p$ Correspondences and Torsion Classes

Recent progress extends compatibility to mod $p$ settings:

• Scholze’s functor: The theory attaches a Weil-equivariant étale sheaf $\mathcal{F}_\pi$ to smooth admissible mod-$p$ representations of $\mathrm{GL}_n$ over $p$-adic fields, and the étale cohomology of $\mathbb{P}^{n-1}_{\mathbb{C}_p}$ equipped with $\mathcal{F}_\pi$ recovers the local residual Galois representation uniquely under flatness and multiplicity-free assumptions (Liu, 2021, Liu et al., 2021).
• Patched module technique for higher rank: Taylor–Wiles patching in the mod $p$ context ties the local factors of automorphic forms to the explicit structure of local Galois representations, handling more complex groups such as $\mathrm{GSp}_4$ (Enns et al., 2021).
• Local–global compatibility in families: The completed cohomology of Shimura varieties at infinite level and ordinary parts yield a compatibly patched module that realizes the local Langlands correspondence in families, even for non-classical and non-étale points (Johansson et al., 2017).

5. Special Models and Generalizations: Kirillov, Jacquet, Hida, and Igusa Varieties

New models are developed to bridge analytic and geometric constructions:

• Completed Kirillov model: For the ordinary locus of Igusa varieties ($\mathrm{GL}_2$), the Banach space completion of the Kirillov model encapsulates $p$-adic automorphic forms, with explicit topological inclusions reflecting the subspace of functions governed by local Langlands data (Howe, 30 Jun 2025).
• Coinvariants and ordinary projectors: Ordinary parts and coinvariants of relevant group actions (e.g., $\widetilde{\mu}_{p^\infty}$) allow identification of classical $p$-adic modular forms with function spaces on Igusa or Mantovan varieties.
• Jacquet–Langlands and boundary induction: Inductive arguments on stratifications of Shimura varieties (Siegel boundary strata, Bruhat filtration) facilitate explicit comparison of Hecke actions and monodromy, leading to the degree-shifting phenomenon essential for compatibility proofs in higher rank (Salazar et al., 13 Aug 2025).
• Global neighborhoods and analytic gluing: In non-arithmetic contexts, compatibility of local geometric neighborhood data (germs) implies the existence of global analytic complexes capturing all local-to-global structures (Coates et al., 2014).

6. Impact, Applications, and Open Problems

Local-global compatibility underpins several central themes in the Langlands program and arithmetic geometry:

• Extraction of arithmetic invariants: Compatibility is essential for relating automorphic $L$-functions to Galois-theoretic and motivic data, including $\mathcal{L}$-invariants arising in exceptional zero phenomena (Salazar et al., 13 Aug 2025, Yang, 29 Jun 2024).
• Construction of eigenvarieties and $p$-adic Langlands: The possibility of interpolating the local Langlands correspondence in $p$-adic analytic families enables the paper of global triangulations, $p$-adic deformation spaces, and potential extensions to the $p$-adic local Langlands program in higher rank (Johansson et al., 2017, Chojecki et al., 2014).
• Mod $p$ and torsion correspondences: The definitive identification of local residual Galois representations from the mod $p$ cohomology of Shimura varieties has ramifications for the mod $p$ Langlands program and modularity lifting (Liu et al., 2021, Liu, 2021, Enns et al., 2021).
• Function field analogues: The uniform geometry of shtuka spaces over function fields extends compatibility to the equal characteristic case, confirming correspondence between global and local parameters (Li-Huerta, 2023).
• Generalizations: Current open questions include full compatibility beyond $\mathrm{GL}_2$, extension to symplectic/orthogonal/shimura settings, understanding overconvergent companions, higher $p$-adic Hodge theory, and the structure of generalized eigenspaces and congruence ideals (Howe, 30 Jun 2025, Hevesi, 2023).

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Table: Local-global Compatibility Results for $\mathrm{GL}_n$ Automorphic Representations

Context (Field/Group)	Compatibility Statement	Citation
$\mathrm{GL}_2$ over CM fields, $\ell=p$	$\mathrm{WD}(r_\iota(\pi)|_{G_{F_v})^{F\text{-ss}} \simeq \mathrm{rec}_{F_v}(\pi_v\otimes|\det|^{-1/2})$	(Yang, 29 Jun 2024)
$\mathrm{GL}_n$ over CM fields, $p\neq l$	Agreement of Frobenius-semisimplified Weil–Deligne parameters	(Matsumoto, 2023, Yang, 2021)
Mod $p$ automorphic forms ($\mathrm{GL}_n$)	$(\Gal_{F^+_{\mathfrak{p}})$-action on the cohomology determines the residual Galois representation	(Liu et al., 2021, Liu, 2021)
Igusa varieties / completed Kirillov model	Weak compatibility of local eigenspaces in $p$-adic Banach automorphic functions	(Howe, 30 Jun 2025)

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Local-global compatibility remains a central mechanism in the arithmetic realization of automorphic forms, confirming the predictive concordance between local representations and global arithmetic data across number fields, function fields, and $p$-adic analytic families. Theoretical advances continue to deepen its reach and refine its statement, bridging analytic, geometric, and cohomological methodologies in modern number theory.