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.