The overconvergence of multivariable $(φ_q,\mathcal{O}_K^{\times})$-modules at the perfectoid level

1. Problem Statement and Context

Background:
The study of $p$-adic Galois representations often utilizes $(\phi, \Gamma)$-modules. Fontaine established an equivalence between continuous representations of the absolute Galois group $G_K$ and étale $(\phi, \Gamma)$-modules.

• Cyclotomic Case: For the cyclotomic extension, Cherbonnier and Colmez proved that these modules are overconvergent, meaning they descend to modules over the Robba ring (Laurent series converging on an open annulus). This is crucial for linking Galois representations to $p$-adic differential equations.
• Lubin-Tate Case: For Lubin-Tate extensions, overconvergence generally fails.
• Multivariable Case: In a previous work ([Du25]), the author introduced multivariable $(\phi_q, \mathcal{O}_K^\times)$-modules over a coefficient ring $A_{mv, E}$ with $f = [K:\mathbb{Q}_p]$ variables. These modules provide a fully faithful functor from $G_K$-representations to modules over $A_{mv, E}$.

The Core Problem:
It remains an open question whether the multivariable $(\phi_q, \mathcal{O}_K^\times)$-modules associated with general Galois representations are overconvergent (i.e., do they descend to a subring $A^\dagger_{mv, E}$ of overconvergent elements?). While the cyclotomic case relies on Tate's normalized trace, this method does not extend easily to the multivariable setting because $\mathcal{O}_K^\times$ is not a one-dimensional $p$-adic Lie group, and the coefficient ring has multiple variables.

Goal:
This paper aims to:

1. Define the category of overconvergent multivariable modules and establish their basic structural properties.
2. Prove overconvergence at the perfectoid level using the geometry of the relative Fargues-Fontaine curve, showing that these modules descend to integral perfectoid Robba rings for arbitrary radii.

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2. Methodology

The paper employs a blend of algebraic module theory and modern $p$-adic geometry.

A. Algebraic Framework: Overconvergent Rings and Adjunctions

• Coefficient Rings: The author defines a subring $A^\dagger_{mv, E} \subset A_{mv, E}$ consisting of overconvergent elements. This is constructed as a union of rings $A^{\dagger, s}_{mv, E}$ defined by convergence conditions on the variables $Y_{\sigma_i}$.
• Functorial Construction:
  • Let $j^{\dagger, *}$ be the base change functor from $A^\dagger_{mv, E}$-modules to $A_{mv, E}$-modules.
  • The author constructs a right adjoint functor $j^\dagger_*$ (following Cherbonnier's thesis) which takes an étale module $M$ over $A_{mv, E}$ and returns the union of its $\phi_q$-stable, finitely generated $A^\dagger_{mv, E}$-submodules.
• Key Algebraic Result: The paper proves that $j^{\dagger, *}$ is fully faithful. A module $M$ is defined as overconvergent if it lies in the essential image of $j^{\dagger, *}$.

B. Geometric Framework: Relative Fargues-Fontaine Curves

To address the perfectoid level, the author utilizes the theory of relative Fargues-Fontaine curves (Scholze-Weinstein, Fargues-Scholze).

• Setup: Consider the perfectoid space $Y_{X, [0, r]}$ associated with a perfectoid algebra $R$. The rings of functions on these spaces are the integral perfectoid Robba rings ($\widetilde{R}^{int}_R$).
• Descent Strategy: Instead of trying to descend directly to the classical overconvergent ring $A^\dagger_{mv, E}$ (which is difficult due to the multi-variable structure), the author descends to the perfectoid ring $\widetilde{R}^{int}_{A_\infty}$.
• Galois Descent via Torsors:
  • The construction involves a map $m: \text{Spa}(A'_\infty) \to \text{Spa}(A_\infty)$ which is a pro-étale $\Delta_1$-torsor.
  • The author uses the fact that vector bundles on the Fargues-Fontaine curve form a $v$-stack (Proposition 1.4).
  • By taking invariants under the group $\Delta_1$ (a subgroup of $(\mathcal{O}_K^\times)^f$), the author constructs modules over the base perfectoid ring from modules over the cover.

C. Technical Lemmas

• Finiteness: Proves that submodules of étale modules over the integral Robba ring are free (Proposition 3.6), leveraging the fact that $\pi$ is in the Jacobson radical.
• Locally Analytic Vectors: Investigates the action of $\mathcal{O}_K^\times$ on the rings, showing that the action on $A^{\dagger, s-}_{mv, E}$ is locally $\mathbb{Q}_p$-analytic (Lemma 3.15), which is a necessary condition for certain descent arguments.

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3. Key Contributions and Results

1. Definition and Structure of Overconvergent Modules (Section 2)

• Definition: Rigorously defines overconvergent multivariable $(\phi_q, \mathcal{O}_K^\times)$-modules.
• Theorem 2.22: Establishes that the functor $j^{\dagger, *}$ is exact and fully faithful. It constructs the right adjoint $j^\dagger_*$ and proves that for any étale module $M$, the natural map $j^{\dagger, *} j^\dagger_* M \to M$ is an isomorphism if $M$ is overconvergent.
• Significance: This provides a robust categorical framework for overconvergence in the multivariable setting, analogous to the cyclotomic case.

2. Perfectoid Overconvergence (Section 3)

• Theorem 1.5 / Theorem 3.9: The main result. For any finite free continuous $\mathcal{O}_E$-representation $\rho$ of $G_K$, and for any radius $r \in \mathbb{Q}_{>0}$, the associated module $D^{[0,r]}_{A_\infty, i}(\rho)$ is a locally free module of rank $d$ over the integral perfectoid Robba ring $B_{A_\infty, [0,r]}$.
• Arbitrary Radius: Unlike previous methods (generalized Colmez-Sen-Tate) which only guaranteed overconvergence for sufficiently small radii, this geometric proof using Fargues-Fontaine curves shows the result holds for arbitrary $r$.
• Proposition 3.10: Shows that for sufficiently small $r$, these modules are actually free (not just locally free).

3. Relation to Classical Rings

• Proposition 3.14: Establishes an embedding $\iota: A^{\dagger, s-}_{mv, E} \hookrightarrow (B_{A_\infty, [0, 1/s]})^\circ$ that preserves norms. This links the algebraic definition of overconvergence to the geometric perfectoid setting.
• Open Question: The paper explicitly notes that while perfectoid overconvergence is proven, descending the result from the perfectoid ring $\widetilde{R}^{int}_{A_\infty}$ back to the classical ring $A^\dagger_{mv, E}$ remains an open problem for general representations.

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4. Significance and Implications

1. Bridging Algebra and Geometry: The paper successfully adapts the powerful machinery of the Fargues-Fontaine curve to the multivariable $(\phi, \Gamma)$-module setting. This demonstrates that geometric methods can overcome algebraic obstructions (like the lack of a simple trace map in the multivariable case).
2. Generalization of Overconvergence: It extends the concept of overconvergence beyond the cyclotomic and Lubin-Tate cases to a broader class of multivariable modules, which are conjectured to be related to $p$-adic representations of $GL_2(K)$.
3. Foundation for Future Work: By proving overconvergence at the perfectoid level, the paper provides a "perfectoid avatar" of these Galois representations. This is a critical step toward understanding the local Langlands correspondence for $GL_n(K)$ in the $p$-adic setting.
4. Methodological Shift: The shift from analytic trace computations (which fail here) to geometric descent via $v$-sheaves and torsors represents a modern approach to $p$-adic Hodge theory problems.

Conclusion

Changjiang Du's paper resolves the overconvergence problem for multivariable $(\phi_q, \mathcal{O}_K^\times)$-modules at the perfectoid level. While the descent to the classical overconvergent ring $A^\dagger_{mv, E}$ for general representations remains open, the proof that these modules are well-behaved over the integral perfectoid Robba rings for arbitrary radii is a significant breakthrough. It utilizes the relative Fargues-Fontaine curve to bypass the limitations of previous analytic methods, offering a new geometric perspective on $p$-adic Galois representations.