Maximal Arboreal Galois Images for Polynomials of Twisted Carlitz Type

Technical Summary: Maximal Arboreal Galois Images for Polynomials of Twisted Carlitz Type

Problem Statement
This paper investigates the arboreal Galois representations associated with polynomials of the form $f_c(x) = cx^d + t$ over the function field $K = \mathbb{F}_q(t)$, where $q = p^e$ is an odd prime power and $d = q-1$. These polynomials are of "twisted Carlitz type," linked to rank-one twisted Carlitz modules via the identity $\rho_c^t(x) = x f_c(x)$.

The central problem is to determine when the image of the arboreal Galois representation, $G_n = \text{Gal}(\text{Spl}_K(f_c^{\circ n})/K)$, achieves "maximality." Specifically, the authors seek conditions under which $G_n$ is isomorphic to the full $n$-fold iterated cyclic wreath product $C_d \wr C_d \wr \cdots \wr C_d$ at every level $n \geq 1$. While such maximality results are well-developed in characteristic zero (often yielding finite index results), the positive characteristic case has historically been obstructed by issues like isotriviality and inseparability, with existing results typically stopping short of full surjectivity.

Methodology
The authors employ a combination of arithmetic dynamics, Kummer theory, and function field arithmetic to establish maximality. The methodology relies on three primary ingredients:

1. Kummer Independence Criterion (Proposition 3.1): The authors reduce the problem of maximality at level $n$ to the linear independence of specific Kummer classes in the field extension generated by the $(n-1)$-th iterates. They prove that if there exists a finite place $P$ of $K$ that "isolates" a single Kummer class (i.e., $v_P(a_n)=1$ while $v_P(a_m)=0$ for $m<n$), then the Galois group at level $n$ is the full wreath product, provided the previous level acts transitively.
2. Primitive Simple Divisor Theorems: To satisfy the valuation condition required by the Kummer criterion, the authors prove the existence of "primitive simple divisors" in the critical orbit $a_n = f_c^{\circ n}(0)$. This is achieved using the Mason–Stothers theorem (the function field analogue of the $abc$ conjecture) over $\mathbb{F}_q[t]$. They analyze the degree growth of the orbit and show that for specific families, the primitive part of the orbit elements must contain irreducible factors with multiplicity one.
3. Capelli's Lemma and Conjugacy: For the second family of polynomials, the authors utilize linear conjugacy to transform $f_c(x) = cx^d + t$ into the simpler form $F(x) = x^d + C$. They use Capelli's Lemma to establish the irreducibility of iterates and the separability of the system, ensuring the Galois groups are well-defined and embed into the expected wreath products.

Key Contributions and Results

The paper establishes two explicit families of polynomials over $\mathbb{F}_q(t)$ that achieve full arboreal maximality at every level, a result previously unattained for explicit families in positive characteristic.

• Main Theorem 1 (Theorem 4.7): For the family $f_M(x) = t^M x^{q-1} + t$, the authors prove that if $M + d - 1 \not\equiv 0 \pmod p$ and the Kummer class $[-t^{1-M}]$ has order $d$, then $G_n \cong C_d \wr \cdots \wr C_d$ ($n$ factors) for all $n \geq 1$. This relies on a primitive simple divisor theorem for the critical orbit of $f_M$.
• Main Theorem 2 (Theorem 5.5): For the family $f_c(x) = cx^{q-1} + t$ where $c \in (K^\times)^{d-1}$ and $v_t(c)=0$, the authors show that the iterated Galois groups are also full cyclic wreath products. This is proven by analyzing the conjugate polynomial $F(x) = x^d + C$ and establishing primitive simple divisors for its critical orbit.

Relationship Between Arboreal and Adelic Representations
A significant portion of the paper (Section 6) compares the arboreal Galois image of $f_c$ with the adelic Galois image of the associated twisted Carlitz module $\rho_c^t$. Using Gekeler's defect formula, the authors demonstrate that:

• Logical Independence: Arboreal maximality and adelic surjectivity are logically independent.
  • They construct an example (Proposition 6.1–6.3) where the adelic image is surjective, but the arboreal image is not maximal (due to the failure of the primitive simple divisor property).
  • They construct a second example (Proposition 6.4) where the arboreal image is maximal, but the adelic image is not surjective (having a non-trivial defect).
• One-Way Local Implication (Main Theorem 3): Despite the general independence, the authors prove a specific local implication at the place $t$. If the first level of the arboreal representation is maximal (i.e., $G_1 \cong C_d$), then the $t$-adic Galois representation of the twisted Carlitz module is surjective (defect is 1). This connection is strictly one-directional and local to the place $t$.

Significance
The paper claims significance by providing the first explicit families in positive characteristic where the arboreal Galois image is the full iterated wreath product at every level, rather than just having finite index. This resolves a gap in the theory of arithmetic dynamics over function fields, which has been hindered by characteristic $p$ obstructions.

Furthermore, the work clarifies the nuanced relationship between polynomial dynamics and Drinfeld module theory. While the first level of the arboreal tower is determined by the torsion of the twisted Carlitz module, the higher towers encode distinct arithmetic information. The paper demonstrates that surjectivity in the Drinfeld module context (adelic) does not guarantee maximality in the polynomial context (arboreal), and vice versa, except for the specific local constraint at the prime $t$.

Finally, the Appendix provides a generalized eventual primitive simple divisor theorem for arbitrary coefficients $c \in \mathbb{F}_q[t]$, suggesting that the phenomenon of primitive divisors in critical orbits is robust and not limited to the specific families analyzed in the main theorems.