[Paper Review] An introduction to the categorical p-adic Langlands program

Matthew Emerton, Toby Gee+1 more|arXiv (Cornell University)|Oct 4, 2022

advanced mathematical theories4 citations

TL;DR

This paper introduces the categorical approach to the $p$-adic Langlands program, unifying Banach and analytic settings through moduli stacks of $(\varphi,\Gamma)$-modules and derived algebraic geometry. It establishes a framework linking $p$-adic Galois representations to automorphic forms via stable $\infty$-categories and derived representation stacks, with key results in $\mathrm{GL}_2(\mathbb{Q}_p)$ and global applications to Shimura variety cohomology.

ABSTRACT

We give an introduction to the "categorical" approach to the p-adic Langlands program, in both the "Banach" and "analytic" settings.

Motivation & Objective

• To develop a categorical framework for the $p$-adic Langlands correspondence, unifying Banach and analytic settings.
• To extend the Taylor–Wiles patching method to derived and categorical structures in $p$-adic Hodge theory.
• To relate Galois representations to automorphic forms through derived moduli stacks and stable $\infty$-categories.
• To compute the cohomology of Shimura varieties using global categorical Langlands functors and eigenvarieties.
• To establish a derived interpretation of group cohomology via the cotangent complex of derived representation stacks.

Proposed method

• Construct moduli stacks of $(\varphi,\Gamma)$-modules over the Robba ring in the analytic setting and over Banach algebras in the Banach setting.
• Use simplicial resolutions of Galois groups to define derived representation stacks $\mathcal{X}_{\Gamma_{\bullet}}$ with cotangent complex $\mathbf{L}_{\mathcal{X}_{\Gamma_{\bullet}},\rho} = C^{\bullet}(\Gamma,\mathrm{Ad}\rho)^*[-1]$.
• Apply derived algebraic geometry to interpret group cohomology in terms of derived tangent spaces and quasi-smoothness.
• Employ stable $\infty$-categories and ind-coherent sheaves on ind-algebraic stacks to formalize the categorical correspondence.
• Utilize Morita theory and pro-coherent sheaves on formal stacks to relate Hecke algebras and global automorphic forms.
• Apply the Fargues–Scholze conjecture and $C$-group constructions to relate $\ell \neq p$ Langlands parameters to $p$-adic objects.

Experimental results

Research questions

• RQ1 How can the $p$-adic Langlands correspondence be reformulated in a categorical, derived framework using $(\varphi,\Gamma)$-modules?
• RQ2 What is the derived structure of the moduli stack of Galois representations, and how does it relate to group cohomology?
• RQ3 How do eigenvarieties and overconvergent $p$-adic automorphic forms arise from categorical Langlands functors?
• RQ4 In what way do derived representation stacks encode the local and global Langlands correspondence for $\mathrm{GL}_2$?
• RQ5 How can the cohomology of Shimura varieties be computed using global moduli stacks of Langlands parameters and derived sheaf theory?

Key findings

• The derived representation stack $\mathcal{X}_{\Gamma_{\bullet}}$ has cotangent complex $\mathbf{L}_{\mathcal{X}_{\Gamma_{\bullet}},\rho} = C^{\bullet}(\Gamma,\mathrm{Ad}\rho)^*[-1]$, linking Galois cohomology to derived geometry.
• For $\mathrm{GL}_2(\mathbb{Q}_p)$, the categorical $p$-adic local Langlands correspondence is realized via semiorthogonal decompositions and derived categories of $(\varphi,\Gamma)$-modules.
• The moduli stack $\mathcal{X}_{\Gamma_{\bullet}}$ is quasi-smooth near $\rho$ when $H^i(\Gamma,\mathrm{Ad}\rho) = 0$ for $i > 2$, implying a local derived complete intersection structure.
• In the global case, the cohomology of $\mathrm{Sh}_{\mathrm{GL}_2, \mathbb{Q}, S}^{\mathrm{odd}}$ is computed via the global-to-local map and eigenvarieties of overconvergent $p$-adic automorphic forms.
• The structure of $\mathcal{X}_{\Gamma_{\bullet}}$ is independent of the choice of simplicial resolution $\Gamma_{\bullet}$, ensuring well-definedness in the $\infty$-categorical framework.
• The framework realizes the $p$-adic Langlands correspondence as a functor between stable $\infty$-categories of sheaves on moduli stacks, with applications to the Fontaine–Mazur conjecture.