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Locally analytic vectors and $\mathbf{Z}_p$ -extensions

This paper disproves Kedlaya's conjecture that locally analytic vectors provide a generalization of $(\varphi,\Gamma)$ -modules for arbitrary infinitely ramified $p$ -adic Lie extensions by demonstrating that, in the anticyclotomic $\mathbf{Z}_p$ -extension setting, the necessary existence of an overconvergent lift of the field of norms fails, thereby precluding the existence of nontrivial locally analytic vectors in the relevant period rings.

Original authors: Léo Poyeton

Published 2026-04-30

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Léo Poyeton

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to understand the hidden architecture of a very complex, infinite building made of numbers. Mathematicians call this building the "Galois group," and it holds the secrets to how numbers behave when you zoom in on a specific prime number, $p$ .

For a long time, mathematicians had a perfect blueprint for one specific type of wing in this building: the "cyclotomic" wing. This blueprint is called a $(\phi, \Gamma)$ -module. It's like a universal translator that turns difficult problems about numbers into easier problems about functions (like polynomials).

The Big Question: Can We Translate the Rest?

Recently, mathematicians wondered: "Can we build this same translator for any wing of the building, even the weird, infinitely ramified ones?"

A famous mathematician, Kedlaya, made a bold guess (a conjecture): Yes. He thought that if you look at the building through a special lens called "locally analytic vectors," you would always find a perfect translator, no matter which wing you were looking at.

The New Lens: "Locally Analytic Vectors"

Think of "locally analytic vectors" as a high-resolution microscope.

In the old days, this microscope only worked if you could dissolve the numbers in a special solvent (inverting $p$ ).
Recently, new tools (developed by Berger, Porat, and others) allowed mathematicians to use this microscope even when the numbers are "dry" (in an integral setting, where $p$ is not dissolved).

The paper by Léo Poyeton asks: If we use this new, dry microscope on a specific type of wing called a $\mathbb{Z}_p$ -extension, what do we see?

The Discovery: Two Possible Worlds

Poyeton looked at these extensions and found that only two things can happen:

The Empty Room: The microscope sees nothing special. The only things you can find are the basic, boring numbers (the ring of integers). In this case, the "translator" doesn't exist.
The Golden Room: The microscope finds a rich, structured room full of special functions. If this happens, it turns out the building behaves exactly like the famous "cyclotomic" wing we already understood.

The Crucial Link: Poyeton proved that finding this "Golden Room" is exactly the same as finding a specific mathematical object called an "overconvergent lift of the field of norms."

Analogy: Imagine the "field of norms" is a shadow cast by the building. An "overconvergent lift" is a 3D model that perfectly recreates that shadow. Poyeton showed that if you can build this 3D model, you get the Golden Room. If you can't, you get the Empty Room.

The Twist: The Anticyclotomic Counterexample

The paper then tests a specific, tricky wing of the building called the anticyclotomic extension (which exists when you are working with a specific type of number field, like the unramified extension of $\mathbb{Q}_p$ of degree 2).

The Hypothesis: If Kedlaya's guess were true, this anticyclotomic wing should have a "Golden Room" (a non-trivial translator).
The Test: Poyeton assumed this room existed and tried to build a specific mathematical object inside it.
The Contradiction: He proved that if this object existed, it would break a different, well-established rule proposed by another mathematician, Laurent Berger. Berger's rule says such an object cannot exist.
The Conclusion: Since the object cannot exist, the "Golden Room" does not exist in the anticyclotomic setting.

The Final Verdict

Because the "Golden Room" is missing in the anticyclotomic setting, Kedlaya's conjecture is false.

What this means: You cannot simply use "locally analytic vectors" to create a universal translator for every infinitely ramified extension. The theory works beautifully for some, but for the anticyclotomic case, the translator breaks down.
The Silver Lining: This doesn't mean the math is broken; it means the theory needs to be more complex. Poyeton suggests that for these tricky cases, we might need a "derived" theory (a more advanced version of the translator that works in multiple dimensions, not just one) to make sense of the anticyclotomic extension.

Summary in One Sentence

This paper uses a new mathematical microscope to show that while a universal theory for number extensions was hoped to exist, a specific, tricky type of extension (the anticyclotomic one) breaks the theory, proving that the "universal translator" conjecture is incorrect.

1. Problem Statement and Context

The paper addresses the generalization of the theory of $(\phi, \Gamma)$ -modules in $p$ -adic Hodge theory.

Classical Setting: Fontaine established an equivalence between $p$ -adic representations of the absolute Galois group $G_K$ and étale $(\phi, \Gamma)$ -modules over the cyclotomic extension $K_{\text{cyc}} = K(\mu_{p^\infty})$ . This theory relies on rings of periods (like $\mathcal{B}^\dagger$ and the Robba ring $\mathcal{R}_K$ ) and the existence of a "lift of the field of norms."
The Generalization Problem: There is a desire to extend this theory to arbitrary infinitely ramified $p$ -adic Lie extensions $K_\infty/K$ .
Kedlaya's Conjecture: Kedlaya conjectured that for any such extension, the theory of locally analytic vectors (introduced by Schneider-Teitelbaum and generalized by Berger-Colmez) within the period rings would provide a robust $(\phi, \Gamma)$ -module theory, even at the integral level.
The Specific Obstacle: While Berger showed that a "finite height" lift of the field of norms restricts the possible extensions (e.g., ruling out the anticyclotomic case), the existence of an overconvergent lift (a weaker condition) was an open question. The paper investigates whether the existence of non-trivial locally analytic vectors in the integral period ring $\tilde{A}^\dagger$ is equivalent to the existence of such an overconvergent lift.

2. Methodology

The author employs a combination of integral $p$ -adic analysis, the theory of fields of norms, and the newly developed theory of integral locally analytic vectors (using Mahler expansions).

Integral Locally Analytic Vectors: The paper utilizes the recent generalization of locally analytic vectors to $\mathbb{Z}_p$ -algebras (by Berger-Rozensztajn, Porat, etc.). Instead of inverting $p$ , these vectors are defined via convergence properties of Mahler expansions (Super-Hölder vectors).
Focus on $\mathbb{Z}_p$ -extensions: The analysis is restricted to totally ramified $\mathbb{Z}_p$ -extensions $K_\infty/K$ . The author assumes $p \neq 2$ .
Structure of $\tilde{A}^\dagger$ : The core technical work involves analyzing the structure of the ring of locally analytic vectors $(\tilde{A}^\dagger_K)^{\text{la}}$ inside the period ring $\tilde{A}^\dagger$ .
The Anticyclotomic Counterexample: The paper constructs a specific scenario where $K = \mathbb{Q}_{p^2}$ $K = Q_{p^{2}}$ and $K_\infty$ $K_{\infty}$ is the anticyclotomic extension. It assumes the existence of a non-trivial locally analytic vector and derives a contradiction using:
- The operator $\nabla$ (Lie derivative of the Galois action).
- Properties of the kernel of the maps $\theta \circ \phi_q^{-n}$ .
- A conjecture by Berger regarding the dimension of eigenspaces in the Robba ring.

3. Key Contributions and Results

A. Dichotomy of Locally Analytic Vectors in $\mathbb{Z}_p$ -Extensions

The paper establishes a fundamental dichotomy for totally ramified $\mathbb{Z}_p$ -extensions:
Theorem 0.1: Let $K_\infty/K$ be a totally ramified $\mathbb{Z}_p$ -extension. Then exactly one of the following holds:

Trivial Case: $(\tilde{A}^\dagger_K)^{\text{la}} = \mathcal{O}_K$ . There are no non-trivial locally analytic vectors.
Cyclotomic-like Case: $(\tilde{A}^\dagger_K)^{\text{la}} = \phi_q^{-\infty}(A^\dagger_K)$ , where $A^\dagger_K$ is a ring of overconvergent functions in one variable. In this case, the structure behaves exactly like the classical cyclotomic setting.

B. Equivalence with Overconvergent Lifts

Theorem 0.2: For a $\mathbb{Z}_p$ -extension, the existence of a non-trivial locally analytic vector in $\tilde{A}^\dagger_K$ is equivalent to the existence of an overconvergent lift of the field of norms.

This result bridges the gap between the analytic properties of the period ring and the arithmetic geometry of the extension.
It implies that if Kedlaya's conjecture were true (requiring non-trivial vectors to encode all representations), then an overconvergent lift must exist for any such extension.

C. Disproof of Kedlaya's Conjecture via the Anticyclotomic Setting

The paper focuses on the anticyclotomic extension of $K = \mathbb{Q}_{p^2}$ .

Assumption: Assume an overconvergent lift exists (implying non-trivial locally analytic vectors exist).
Construction: Using the existence of a specific element invariant under Frobenius $\phi_q$ with locally analytic Galois action, the author constructs an element in the Robba ring $\mathcal{R}_K$ .
Contradiction: This construction leads to the existence of a non-trivial element in $(\text{Frac}\mathcal{R}_K)^{\phi_q=1}$ (specifically, an eigenspace of dimension $>1$ ).
Berger's Conjecture: This contradicts a conjecture by Berger (Conjecture A in [Ber22]), which states that such eigenspaces should have dimension at most 1.
Conclusion: Since Berger's conjecture is proven to hold in this specific setting (via the properties of the constructed elements), the initial assumption must be false.

Theorem 0.3: There is no overconvergent lift of the field of norms in the anticyclotomic setting. Consequently, Kedlaya's conjecture is false in general.

D. Higher Locally Analytic Vectors

The paper concludes that in cases where $(\tilde{A}^\dagger_K)^{\text{la}} = \mathcal{O}_K$ (the trivial case), the theory of $(\phi, \Gamma)$ -modules cannot be concentrated in degree 0. This necessitates the use of derived $(\phi, \Gamma)$ -modules (higher locally analytic vectors) to capture the representation theory, explaining why the anticyclotomic theory is more complex than the cyclotomic one.

4. Significance

Refutation of a Major Conjecture: The paper provides a definitive counterexample to Kedlaya's conjecture that locally analytic vectors provide a uniform $(\phi, \Gamma)$ -module theory for all infinitely ramified $p$ -adic Lie extensions. It shows that the theory fails for the anticyclotomic extension.
Characterization of Extensions: It clarifies that the existence of a "nice" $(\phi, \Gamma)$ -module theory (via locally analytic vectors) is strictly tied to the existence of an overconvergent lift of the field of norms.
Integral Theory: The work successfully applies the new integral theory of locally analytic vectors (Super-Hölder vectors) to solve deep arithmetic problems, demonstrating the power of this framework in characteristic $p$ and integral settings.
Future Directions: The results suggest that for extensions like the anticyclotomic one, one must look beyond degree 0 and utilize derived categories (higher locally analytic vectors) to formulate a correct $p$ -adic Hodge theory.

In summary, Poyeton proves that the "naive" generalization of $(\phi, \Gamma)$ -modules via locally analytic vectors fails for anticyclotomic extensions because the necessary overconvergent lifts of the field of norms do not exist, thereby disproving Kedlaya's conjecture in its general form.