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In arithmetic dynamics, an arboreal Galois representation is a continuous group homomorphism between the absolute Galois group of a field and the automorphism group of an infinite, regular, rooted tree.

The study of arboreal Galois representations of goes back to the works of Odoni in 1980s.

Definition

Let ${\displaystyle K}$ be a field and ${\displaystyle K^{sep}}$ be its separable closure. The Galois group ${\displaystyle G_{K}}$ of the extension ${\displaystyle K^{sep}/K}$ is called the absolute Galois group of ${\displaystyle K}$. This is a profinite group and it is therefore endowed with its natural Krull topology.

For a positive integer ${\displaystyle d}$, let ${\displaystyle T^{d}}$ be the infinite regular rooted tree of degree ${\displaystyle d}$. This is an infinite tree where one node is labeled as the root of the tree and every node has exactly ${\displaystyle d}$ descendants. An automorphism of ${\displaystyle T^{d}}$ is a bijection of the set of nodes that preserves vertex-edge connectivity. The group ${\displaystyle Aut(T^{d})}$ of all automorphisms of ${\displaystyle T^{d}}$ is a profinite group as well, as it can be seen as the inverse limit of the automorphism groups of the finite sub-trees ${\displaystyle T_{n}^{d}}$ formed by all nodes at distance at most ${\displaystyle n}$ from the root. The group of automorphisms of ${\displaystyle T_{n}^{d}}$ is isomorphic to ${\displaystyle S_{d}\wr S_{d}\wr \ldots \wr S_{d}}$, the iterated wreath product of ${\displaystyle n}$ copies of the symmetric group of degree ${\displaystyle d}$.

An arboreal Galois representation is a continuous group homomorphism ${\displaystyle G_{K}\to Aut(T^{d})}$.

Arboreal Galois representations attached to rational functions

The most natural source of arboreal Galois representations is the theory of iterations of self-rational functions on the projective line. Let ${\displaystyle K}$ be a field and ${\displaystyle f\colon \mathbb {P} _{K}^{1}\to \mathbb {P} _{K}^{1}}$ a rational function of degree ${\displaystyle d}$. For every ${\displaystyle n\geq 1}$ let ${\displaystyle f^{n}=f\circ f\circ \ldots \circ f}$ be the ${\displaystyle n}$-fold composition of the map ${\displaystyle f}$ with itself. Let ${\displaystyle \alpha \in K}$ and suppose that for every ${\displaystyle n\geq 1}$ the set ${\displaystyle (f^{n})^{-1}(\alpha )}$ contains ${\displaystyle d^{n}}$ elements of the algebraic closure ${\displaystyle {\overline {K}}}$. Then one can construct an infinite, regular, rooted ${\displaystyle d}$-ary tree ${\displaystyle T(f)}$ in the following way: the root of the tree is ${\displaystyle \alpha }$, and the nodes at distance ${\displaystyle n}$ from ${\displaystyle \alpha }$ are the elements of ${\displaystyle (f^{n})^{-1}(\alpha )}$. A node ${\displaystyle \beta }$ at distance ${\displaystyle n}$ from ${\displaystyle \alpha }$ is connected with an edge to a node ${\displaystyle \gamma }$ at distance ${\displaystyle n+1}$ from ${\displaystyle \alpha }$ if and only if ${\displaystyle f(\beta )=\gamma }$.

The absolute Galois group ${\displaystyle G_{K}}$ acts on ${\displaystyle T(f)}$ via automorphisms, and the induced homorphism ${\displaystyle \rho _{f,\alpha }\colon G_{K}\to Aut(T(f))}$ is continuous, and therefore is called the arboreal Galois representation attached to ${\displaystyle f}$ with basepoint ${\displaystyle \alpha }$.

Arboreal representations attached to rational functions can be seen as a wide generalization of Galois representations on Tate modules of abelian varieties.

Arboreal Galois representations attached to quadratic polynomials

The simplest non-trivial case is that of monic quadratic polynomials. Let ${\displaystyle K}$ be a field of characteristic not 2, let ${\displaystyle f=(x-a)^{2}+b\in K}$ and set the basepoint ${\displaystyle \alpha =0}$. The adjusted post-critical orbit of ${\displaystyle f}$ is the sequence defined by ${\displaystyle c_{1}=-f(a)}$ and ${\displaystyle c_{n}=f^{n}(a)}$ for every ${\displaystyle n\geq 2}$. A resultant argument shows that ${\displaystyle (f^{n})^{-1}(0)}$ has ${\displaystyle d^{n}}$ elements for ever ${\displaystyle n}$ if and only if ${\displaystyle c_{n}\neq 0}$ for every ${\displaystyle n}$. In 1992, Stoll proved the following theorem:

Theorem: the arboreal representation ${\displaystyle \rho _{f,0}}$ is surjective if and only if the span of ${\displaystyle \{c_{1},\ldots ,c_{n}\}}$ in the ${\displaystyle \mathbb {F} _{2}}$-vector space ${\displaystyle K^{*}/(K^{*})^{2}}$ is ${\displaystyle n}$-dimensional for every ${\displaystyle n\geq 1}$.

The following are examples of polynomials that satisfy the conditions of Stoll's Theorem, and that therefore have surjective arboreal representations.

• For ${\displaystyle K=\mathbb {Q} }$, ${\displaystyle f=x^{2}+a}$, where ${\displaystyle a\in \mathbb {Z} }$ is such that either ${\displaystyle a>0}$ and ${\displaystyle a\equiv 1,2{\bmod {4}}}$ or ${\displaystyle a<0}$, ${\displaystyle a\equiv 0{\bmod {4}}}$ and ${\displaystyle -a}$ is not a square.

• Let ${\displaystyle k}$ be a field of characteristic not ${\displaystyle 2}$ and ${\displaystyle K=k(t)}$ be the rational function field over ${\displaystyle k}$. Then ${\displaystyle f=x^{2}+t\in K}$ has surjective arboreal representation.

Higher degrees and Odoni's conjecture

In 1985 Odoni formulated the following conjecture.

Conjecture: Let ${\displaystyle K}$ be a Hilbertian field of characteristic ${\displaystyle 0}$, and let ${\displaystyle n}$ be a positive integer. Then there exists a polynomial ${\displaystyle f\in K}$ of degree ${\displaystyle n}$ such that ${\displaystyle \rho _{f,0}}$ is surjective.

Although in this very general form the conjecture has been shown to be false by Dittmann and Kadets, there are several results when ${\displaystyle K}$ is a number field. Benedetto and Juul proved Odoni's conjecture for ${\displaystyle K}$ a number field and ${\displaystyle n}$ even, and also when both ${\displaystyle }$ and ${\displaystyle n}$ are odd, Looper independently proved Odoni's conjecture for ${\displaystyle n}$ prime and ${\displaystyle K=\mathbb {Q} }$.

Finite index conjecture

When ${\displaystyle K}$ is a global field and ${\displaystyle f\in K(x)}$ is a rational function of degree 2, the image of ${\displaystyle \rho _{f,0}}$ is expected to be "large" in most cases. The following conjecture quantifies the previous statement, and it was formulated by Jones in 2013.

Conjecture Let ${\displaystyle K}$ be a global field and ${\displaystyle f\in K(x)}$ a rational function of degree 2. Let ${\displaystyle \gamma _{1},\gamma _{2}\in \mathbb {P} _{K}^{1}}$ be the critical points of ${\displaystyle f}$. Then ${\displaystyle =\infty }$ if and only if at least one of the following conditions hold:

1. The map ${\displaystyle f}$ is post-critically finite, namely the orbits of ${\displaystyle \gamma _{1},\gamma _{2}}$ are both finite.
2. There exists ${\displaystyle n\geq 1}$ such that ${\displaystyle f^{n}(\gamma _{1})=f^{n}(\gamma _{2})}$.
3. ${\displaystyle 0}$ is a periodic point for ${\displaystyle f}$.
4. There exist a Möbius transformation ${\displaystyle m={\frac {ax+b}{cx+d}}\in PGL_{2}(K)}$ that fixes ${\displaystyle 0}$ and is such that ${\displaystyle m\circ f\circ m^{-1}=f}$.

Jones' conjecture is considered to be a dynamical analogue of Serre's open image theorem.

One direction of Jones' conjecture is known to be true: if ${\displaystyle f}$ satisfies one of the above conditions, then ${\displaystyle =\infty }$. In particular, when ${\displaystyle f}$ is post-critically finite then ${\displaystyle Im(\rho _{f,\alpha })}$ is a topologically finitely generated closed subgroup of ${\displaystyle Aut(T(f))}$ for every ${\displaystyle \alpha \in K}$.

In the other direction, Juul et al. proved that if the abc conjecture holds for number fields, ${\displaystyle K}$ is a number field and ${\displaystyle f\in K}$ is a quadratic polynomial, then ${\displaystyle =\infty }$ if and only if ${\displaystyle f}$ is post-critically finite or not eventually stable. When ${\displaystyle f\in K}$ is a quadratic polynomial, conditions (2) and (4) in Jones' conjecture are never satisfied. Moreover, Jones and Levy conjectured that ${\displaystyle f}$ is eventually stable if and only if ${\displaystyle 0}$ is not periodic for ${\displaystyle f}$.

Abelian arboreal representations

In 2020, Andrews and Petsche formulated the following conjecture.

Conjecture Let ${\displaystyle K}$ be a number field, let ${\displaystyle f\in K}$ be a polynomial of degree ${\displaystyle d\geq 2}$ and let ${\displaystyle \alpha \in K}$. Then ${\displaystyle Im(\rho _{f,\alpha })}$ is abelian if and only if there exists a root of unity ${\displaystyle \zeta }$ such that the pair ${\displaystyle (f,\alpha )}$ is conjugate over the maximal abelian extension ${\displaystyle K^{ab}}$ to ${\displaystyle (x^{d},\zeta )}$ or to ${\displaystyle (\pm T_{d},\zeta +\zeta ^{-1})}$, where ${\displaystyle T_{d}}$ is the Chebyshev polynomial of the first kind of degree ${\displaystyle d}$.

Two pairs ${\displaystyle (f,\alpha ),(g,\beta )}$, where ${\displaystyle f,g\in K(x)}$ and ${\displaystyle \alpha ,\beta \in K}$ are conjugate over a field extension ${\displaystyle L/K}$ if there exists a Möbius transformation ${\displaystyle m={\frac {ax+b}{cx+d}}\in PGL_{2}(L)}$ such that ${\displaystyle m\circ f\circ m^{-1}=g}$ and ${\displaystyle m(\alpha )=\beta }$. Conjugacy is an equivalence relation. The Chebyshev polynomials the conjecture refers to are a normalized version, conjugate by the Möbius transformation ${\displaystyle 2x}$ to make them monic.

It has been proven that Andrews and Petsche's conjecture holds true when ${\displaystyle K=\mathbb {Q} }$.

References

1. Jones, Rafe (2008). "The density of prime divisors in the arithmetic dynamics of quadratic polynomials". Journal of the London Mathematical Society. 78 (2): 523–544. arXiv:math/0612415. doi:10.1112/jlms/jdn034. S2CID 15310955.
2. ^ Stoll, Michael (1992). "Galois groups over ${\displaystyle cQ}$ of some iterated polynomials". Archiv der Mathematik. 59 (3): 239–244. doi:10.1007/BF01197321. S2CID 122514918.
3. Ferraguti, Andrea; Micheli, Giacomo (2020). "An equivariant isomorphism theorem for mod ${\displaystyle {\mathfrak {p}}}$ reductions of arboreal Galois representations". Trans. Amer. Math. Soc. 373 (12): 8525–8542. arXiv:1905.00506. doi:10.1090/tran/8247.
4. Odoni, R.W.K. (1985). "The Galois theory of iterates and composites of polynomials". Proceedings of the London Mathematical Society. 51 (3): 385–414. doi:10.1112/plms/s3-51.3.385.
5. Dittmann, Philip; Kadets, Borys (2022). "Odoni's conjecture on arboreal Galois representations is false". Proc. Amer. Math. Soc. 150 (8): 3335–3343. arXiv:2012.03076. doi:10.1090/proc/15920.
6. Benedetto, Robert; Juul, Jamie (2019). "Odoni's conjecture for number fields". Bulletin of the London Mathematical Society. 51 (2): 237–250. arXiv:1803.01987. doi:10.1112/blms.12225. S2CID 53400216.
7. Looper, Nicole (2019). "Dynamical Galois groups of trinomials and Odoni's conjecture". Bulletin of the London Mathematical Society. 51 (2): 278–292. arXiv:1609.03398. doi:10.1112/blms.12227.
8. Jones, Rafe (2013). Galois representations from pre-image trees: an arboreal survey. Actes de la Conférence Théorie des Nombres et Applications. pp. 107–136.
9. Jones, Rafe; Levy, Alon (2017). "Eventually stable rational functions". Int. J. Number Theory. 13 (9): 2299–2318. arXiv:1603.00673. doi:10.1142/S1793042117501263. S2CID 119704204.
10. Andrews, Jesse; Petsche, Clayton (2020). "Abelian extensions in dynamical Galois theory". Algebra Number Theory. 14 (7): 1981–1999. arXiv:2001.00659. doi:10.2140/ant.2020.14.1981. S2CID 209832399.
11. Ferraguti, Andrea; Ostafe, Alina; Zannier, Umberto (2024). "Cyclotomic and abelian points in backward orbits of rational functions". Advances in Mathematics. 438. arXiv:2203.10034. doi:10.1016/j.aim.2023.109463. S2CID 247594240.

Further reading

• Arboreal Galois Representations Over Finite Fields
• Li, Hua-Chieh (4 October 2019). "Arboreal Galois representation for a certain type of quadratic polynomials". Archiv der Mathematik. 114 (3): 265–269. doi:10.1007/s00013-019-01390-x.
• https://www.quora.com/What-is-the-significance-of-arboreal-Galois-representations

• Arithmetic dynamics
• Galois theory