Lubin's conjecture for height-one $p$-adic dynamical systems over $(p^2-p)$-tame extensions

This paper proves Lubin's conjecture in new cases by demonstrating that over $(p^2-p)$-tame extensions, the consistent sequences associated with a commuting pair of noninvertible and invertible height-one formal power series form a crystalline character of weight 1.

The Gist

Imagine you are a detective trying to solve a mystery in a very strange, mathematical city called $p$-adic Land.

In this city, there are two special characters, F and U.

• F is a "shrinker." It takes numbers and squishes them down (it's non-invertible).
• U is a "rotator." It spins numbers around without squishing them (it's invertible).

The mystery begins because F and U are best friends who always commute. No matter who goes first, the result is the same: $F(U(x)) = U(F(x))$.

The Big Question (Lubin's Conjecture)

Back in the 1990s, a mathematician named Lubin asked a simple but deep question:

"If F and U are such good friends that they commute, does that mean they are both employees of a hidden, secret organization called a Formal Group?"

A Formal Group is like a hidden rulebook or a secret society that governs how numbers interact. If F and U are employees, they must follow the society's specific laws. Lubin suspected that any time you see these two types of functions commuting, a secret society must be hiding in the background.

The Detective's Challenge

For a long time, mathematicians could only solve this mystery in very specific, easy neighborhoods of the city (like the integers $\mathbb{Z}_p$). But the city is huge, and in some neighborhoods, the ground is "slippery" (highly ramified). The rules get messy, and the secret society is hard to find.

This paper, written by Martin DeBaisieux, solves the mystery for a specific, tricky neighborhood: places where the "slippery factor" (ramification) isn't too crazy (specifically, not divisible by $p^2 - p$).

How the Detective Solved It (The Analogy)

Here is the step-by-step logic of the paper, translated into everyday terms:

1. The "Consistent Sequence" Trail

The detective starts by looking at the "footprints" left by F.

• Imagine F leaves a trail of footprints: $x_0, x_1, x_2...$ where $F(x_1) = x_0$, $F(x_2) = x_1$, and so on.
• The detective collects all possible infinite trails that fit this pattern. This collection is called the Tate Module.
• In a perfect world (if Lubin is right), this collection of footprints should behave exactly like a secret code (a mathematical character) that reveals the existence of the hidden Formal Group.

2. The "Galois" Security Guard

In $p$-adic Land, there is a security guard called the Galois Group. This guard watches over all the numbers and rearranges them in specific ways.

• The detective notices that the footprints (the Tate Module) move around in a very organized way when the guard rearranges things.
• The detective realizes: "Aha! This movement isn't random. It follows a strict pattern, like a dance."
• By analyzing how the "rotator" U moves these footprints, the detective can decode the pattern. This pattern turns out to be a crystalline character.

3. The "Crystal" Connection

Think of a crystalline character as a piece of frozen data or a perfect crystal.

• In the world of $p$-adic math, if you find a "crystal" of a certain type (weight 1), it is a guarantee that a Formal Group exists. It's like finding a specific type of fossil that proves a dinosaur once lived there.
• The detective proves that the footprints of F and U form this perfect crystal.
• The Catch: The detective had to restrict the view to a slightly smaller neighborhood (a finite extension $L$) to see the crystal clearly because the ground was too slippery in the original neighborhood.

4. Rebuilding the Secret Society

Now that the detective has the crystal (the Tate Module structure), they need to rebuild the secret society (the Formal Group) from scratch.

• They use a high-tech reconstruction kit called Integral $p$-adic Hodge Theory.
• Think of this kit as a 3D printer. You feed it the "frozen data" (the crystal), and it prints out the Formal Group.
• The detective carefully tracks F and U through every step of the printing process to ensure they are still employees of this new group.

5. The Final Verdict

The printer finishes, and out comes a Formal Group defined over the original ring of integers.

• The detective checks: "Do F and U work here?"
• Yes! They are both endomorphisms (employees) of this group.
• Because the group is unique, it must be the same group that was hiding in the background all along.

The Conclusion

The paper proves that for a wide range of "slippery" neighborhoods, if you see a shirker (F) and a rotator (U) commuting, there is definitely a hidden Formal Group governing their relationship.

In short:
The paper takes a complex mathematical puzzle about how two functions interact, uses the "footprints" they leave behind to find a hidden "crystal" pattern, and then uses that pattern to reconstruct the secret organization that binds them together. It confirms Lubin's hunch: Commuting implies a hidden structure.

Technical Summary

Here is a detailed technical summary of the paper "Lubin's Conjecture for Height-One p-Adic Dynamical Systems over $(p^2 - p)$-Tame Extensions" by Martin Debaisieux.

1. Problem Statement

The paper addresses Lubin's Conjecture (1994) regarding $p$-adic dynamical systems. The conjecture posits that if a non-invertible formal power series $f$ and a non-torsion invertible formal power series $u$ commute ($f \circ u = u \circ f$) over the ring of integers $\mathcal{O}_K$ of a finite extension $K/\mathbb{Q}_p$, and satisfy specific conditions (simple roots for iterates of $f$, $f'(0)$ is a uniformizer), then there exists a formal group $F$ over $\mathcal{O}_K$ such that both $f$ and $u$ are endomorphisms of $F$.

Specific Context of this Paper:

• Height: The paper focuses on the height-1 case.
• Field Extension: It considers extensions $K/\mathbb{Q}_p$ where the ramification index $e_K$ satisfies the condition $(e_K, p^2 - p) = 1$. This is referred to as the $(p^2 - p)$-tame case.
• Previous Work: The height-1 case over $\mathbb{Z}_p$ was solved by Specter (2018) using $p$-adic analysis. Berger (2019) solved all finite height cases over $\mathbb{Z}_p$ using $p$-adic analysis. This paper extends the result to ramified extensions using integral $p$-adic Hodge theory.
• Assumption: The author assumes the linear coefficient of the invertible series lies in $\mathbb{Z}_p^\times$ (i.e., $u'(0) \in \mathbb{Z}_p^\times$), a condition the author notes is reasonable but not yet proven to follow from the original hypotheses.

2. Methodology

The author employs a strategy of reconstruction via Galois representations, moving from dynamical data to formal group structures through the lens of integral $p$-adic Hodge theory. The methodology proceeds in three main stages:

A. Analysis of the Dynamical System (Section 1)

• Newton Polygons: The author analyzes the roots of the iterates $f^{\circ n}$ using Newton polygons. Under the condition $(e, p^2-p)=1$, it is shown that the Weierstrass polynomials associated with the roots of $f^{\circ n+1}/f^{\circ n}$ are irreducible over $K$.
• Consistent Sequences: The set $T_f$ of $f$-consistent sequences (sequences $(x_n)$ such that $f(x_{n+1}) = x_n$) is identified as the candidate for the $p$-adic Tate module of the latent formal group.
• Galois Action: The action of the absolute Galois group $G_K$ on $T_f$ is studied. By restricting to a finite extension $L$ (generated by roots of specific iterates), the author constructs a character $\chi_f: G_L \to \mathbb{Z}_p^\times$ describing the Galois action on "cells" of $T_f$ via the iterates of $u$.

B. Construction of a Crystalline Character (Section 2)

• Period Rings: The author utilizes Fontaine's period rings ($B_{dR}$ and $B_{cris}$) to analyze the character $\chi_f$.
• Universal Ring: A universal $f^{\circ r}$-consistent ring $A_\infty(\mathcal{O}_L)$ is constructed. By mapping this ring into the tilt of $\mathbb{C}_p$ (via the ring $R$), the author constructs a specific period $t_f = \text{Log}_f(x_0)$.
• Weight Calculation: It is proven that the character $\chi_f$ is crystalline of Hodge-Tate weight 1. This is a crucial step, as crystalline characters of weight 1 correspond to formal groups of height 1.
• Transport of Structure: Using the equivalence between crystalline representations and formal groups, the author identifies a "pivotal" formal group $H$ over $L$ whose Tate module $T_p(H)$ is isomorphic to the character $\chi_f$. The structure of $T_p(H)$ is then transported back to the set $T_f$, endowing $T_f$ with a $\mathbb{Z}_p[G_L]$-module structure where $f$ acts as an endomorphism.

C. Reconstruction of the Formal Group (Section 3)

• Integral $p$-adic Hodge Theory: To recover the formal group over the integers (not just up to isomorphism), the author uses the functors of Kisin and Zink.
  1. Breuil-Kisin Module: The set $T_f$ is converted into a formal Breuil-Kisin module $M_f$ over the Kisin ring $S$.
  2. S-Window: $M_f$ is extended to an $S$-window $W_f$.
  3. $p$-Divisible Group: Using Zink's theory of displays, the window is transformed into a connected $p$-divisible group $\Gamma_f$ over $\mathcal{O}_L$.
• Descent: The formal group law associated with $\Gamma_f$ is shown to be defined over $\mathcal{O}_K$ (the descent from $L$ to $K$) due to the uniqueness of the formal group associated with the specific endomorphism $f$.

3. Key Contributions

1. Generalization to Ramified Extensions: The paper solves Lubin's conjecture for height-1 systems over extensions $K/\mathbb{Q}_p$ where the ramification index is coprime to $p^2 - p$. This covers cases where the ramification is non-trivial but "tame" relative to $p^2-p$.
2. Integration of Hodge Theory: Unlike previous solutions over $\mathbb{Z}_p$ which relied on $p$-adic analysis (Newton polygons and logarithms), this work successfully applies integral $p$-adic Hodge theory (Kisin modules and Zink displays) to reconstruct the formal group from the dynamical system.
3. Recovery of the Latent Formal Group: The paper provides a constructive proof that the set of consistent sequences $T_f$ is not just a Galois set, but carries a canonical $\mathbb{Z}_p$-module structure that makes it isomorphic to the Tate module of a formal group.
4. Handling Non-Eisenstein Polynomials: The author addresses the technical difficulty that, in the ramified case, the Weierstrass polynomials of the iterates are not necessarily Eisenstein, yet remain irreducible, allowing for the transitive Galois action required for the construction.

4. Main Results

Theorem 3.1 (Main Theorem):
Let $K/\mathbb{Q}_p$ be a finite extension with ramification index $e_K$ such that $(e_K, p^2 - p) = 1$. Let $(f, u)$ be a commuting pair in $S_0(\mathcal{O}_K)$ where:

• $f$ has exactly $p$ roots in the open unit disk, all iterates have simple roots, and $f'(0)$ is a uniformizer in $\mathbb{Z}_p$.
• $u$ is non-torsion, invertible, and $u'(0) \in \mathbb{Z}_p^\times$.

Conclusion: There exists a formal group $F$ over $\mathcal{O}_K$ such that $f, u \in \text{End}_{\mathcal{O}_K}(F)$.

5. Significance

• Advancement of Lubin's Conjecture: This work resolves a significant open case of Lubin's conjecture, moving beyond the unramified case ($\mathbb{Z}_p$) to a class of ramified extensions.
• Methodological Bridge: It demonstrates the power of integral $p$-adic Hodge theory in solving problems in $p$-adic dynamics. It bridges the gap between the "dynamical" definition of consistent sequences and the "arithmetic" definition of formal groups via the equivalence of categories between crystalline representations and formal groups.
• Foundation for Future Work: By establishing the framework for height-1 systems in ramified extensions, this paper provides a potential pathway for tackling the remaining open cases of Lubin's conjecture (higher heights or more wildly ramified extensions), although the author notes the remaining cases are still open.

In summary, Debaisieux proves that the algebraic structure implied by the commutativity of $f$ and $u$ in these specific ramified settings is sufficient to guarantee the existence of an underlying formal group, utilizing deep tools from modern arithmetic geometry to reconstruct that group explicitly.