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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.

Definition

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Let be a field and be its separable closure. The Galois group of the extension is called the absolute Galois group of . This is a profinite group and it is therefore endowed with its natural Krull topology.

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

An arboreal Galois representation is a continuous group homomorphism .

Arboreal Galois representations attached to rational functions

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The most natural source of arboreal Galois representations is the theory of iterations of self-rational functions on the projective line. Let be a field and a rational function of degree . For every let be the -fold composition of the map with itself. Let and suppose that for every the set contains elements of the algebraic closure . Then one can construct an infinite, regular, rooted -ary tree in the following way: the root of the tree is , and the nodes at distance from are the elements of . A node at distance from is connected with an edge to a node at distance from if and only if .

The first three levels of the tree of preimages of under the map

The absolute Galois group acts on via automorphisms, and the induced homorphism is continuous, and therefore is called the arboreal Galois representation attached to with basepoint .

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

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The simplest non-trivial case is that of monic quadratic polynomials. Let be a field of characteristic not 2, let and set the basepoint . The adjusted post-critical orbit of is the sequence defined by and for every . A resultant argument[1] shows that has elements for ever if and only if for every . In 1992, Stoll proved the following theorem:[2]

Theorem: the arboreal representation is surjective if and only if the span of in the -vector space is -dimensional for every .

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

  • For , , where is such that either and or , and is not a square.[3]
  • Let be a field of characteristic not and be the rational function field over . Then has surjective arboreal representation[4].

Higher degrees and Odoni's conjecture

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In 1985 Odoni formulated the following conjecture.[5]

Conjecture: Let be a Hilbertian field of characteristic , and let be a positive integer. Then there exists a polynomial of degree such that is surjective.

Although in this very general form the conjecture has been shown to be false by Dittmann and Kadets[6], there are several results when is a number field. Benedetto and Juul proved Odoni's conjecture for a number field and even, and also when both and are odd[7], Looper independently proved Odoni's conjecture for prime and [8], and in an unpublished paper SpecterCite error: There are <ref> tags on this page without content in them (see the help page). layed out a proof of Odoni's conjecture for a number field and any positive integer[9].

Finite index conjecture

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When is a global field and is a rational function of degree 2, the image of is expected to be "large" in most cases. The following conjecture quantifies the previous statement, and it was formulated by Jones in 2013.[10]

Conjecture Let be a global field and a rational function of degree 2. Let be the critical points of . Then if and only if at least one of the following conditions hold:

(1) The map is post-critically finite, namely the orbits of are both finite.

(2) There exists such that .

(3) is a periodic point for .

(4) There exist a Möbius transformation that fixes and is such that .

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 satisfies one of the above conditions, then . In particular, when is post-critically finite then is a topologically finitely generated closed subgroup of for every .

In the other direction, Juul et al. proved that if the abc conjecture holds for number fields, is a number field and is a quadratic polynomial, then if and only if is post-critically finite or not eventually stable. When is a quadratic polynomial, conditions (2) and (4) in Jones' conjecture are never satisfied. Moreover, Jones and Levy conjectured that is eventually stable if and only if is not periodic for [11].

Abelian arboreal representations

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In 2020, Andrews and Petsche formulated the following conjecture[12].

Conjecture Let be a number field, let be a polynomial of degree and let . Then is abelian if and only if there exists a root of unity such that the pair is conjugate over the maximal abelian extension to or to , where is the Chebyshev polynomial of the first kind of degree .

Two pairs , where and are conjugate over a field extension if there exists a Möbius transformation such that and . Conjugacy is an equivalence relation. The Chebyshev polynomials the conjecture refers to are a normalized version, conjugate by the Möbius transformation to make them monic.

It has been proven that Andrews and Petsche's conjecture holds true when [13].

References

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  1. ^ Jones, Rafe (2008). "The density of prime divisors in the arithmetic dynamics of quadratic polynomials". J. Lond. Math. Soc. (2). 78 (2): 523–544.
  2. ^ Stoll, Michael (1992). "Galois groups over of some iterated polynomials". Arch. Math. (Basel). 59 (3): 239–244.
  3. ^ Stoll, Michael (1992). "Galois groups over of some iterated polynomials". Arch. Math. (Basel). 59 (3): 239–244.
  4. ^ Ferraguti, Andrea; Micheli, Giacomo (2020). "An equivariant isomorphism theorem for mod reductions of arboreal Galois representations". Trans. Amer. Math. Soc. 373 (12): 8525–8542.
  5. ^ Odoni, R.W.K. (1985). "The Galois theory of iterates and composites of polynomials". Proc. London Math. Soc. (3). 51 (3): 385–414.
  6. ^ Dittmann, Philip; Kadets, Borys (2022). "Odoni's conjecture on arboreal Galois representations is false". Proc. Amer. Math. Soc. 150 (8): 3335–3343.
  7. ^ Benedetto, Robert; Juul, Jamie (2019). "Odoni's conjecture for number fields". Bull. Lond. Math. Soc. 51 (2): 237–250.
  8. ^ Looper, Nicole (2019). "Dynamical Galois groups of trinomials and Odoni's conjecture". Bull. Lond. Math. Soc. 51 (2): 278–292.
  9. ^ Specter, Joel (2018). "Polynomials with Surjective Arboreal Galois Representations Exist in Every Degree". arXiv:1803.00434.
  10. ^ Jones, Rafe (2013). Galois representations from pre-image trees: an arboreal survey. Actes de la Conférence "Théorie des Nombres et Applications. pp. 107–136.
  11. ^ Jones, Rafe; Levy, Alon (2017). "Eventually stable rational functions". Int. J. Number Theory. 13 (9): 2299–2318.
  12. ^ Andrews, Jesse; Petsche, Clayton (2020). "Abelian extensions in dynamical Galois theory". Algebra Number Theory. 14 (7): 1981–1999.
  13. ^ Ferraguti, Andrea; Ostafe, Alina; Zannier, Umberto (2024). "Cyclotomic and abelian points in backward orbits of rational functions". Adv. Math. 438.