Galois representations from pre-image trees: an arboreal survey

@article{Jones2014GaloisRF,
  title={Galois representations from pre-image trees: an arboreal survey},
  author={Rafe Jones},
  journal={arXiv: Number Theory},
  year={2014}
}
  • Rafe Jones
  • Published 2014
  • Mathematics
  • arXiv: Number Theory
Given a global field K and a rational function phi defined over K, one may take pre-images of 0 under successive iterates of phi, and thus obtain an infinite rooted tree T by assigning edges according to the action of phi. The absolute Galois group of K acts on T by tree automorphisms, giving a subgroup G(phi) of the group Aut(T) of all tree automorphisms. Beginning in the 1980s with work of Odoni, and developing especially over the past decade, a significant body of work has emerged on the… Expand
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References

SHOWING 1-10 OF 62 REFERENCES
Galois theory of quadratic rational functions
For a number field K with absolute Galois group G_K, we consider the action of G_K on the infinite tree of preimages of a point in K under a degree-two rational function phi, with particularExpand
Iterated Galois towers, their associated martingales, and the $p$-adic Mandelbrot set
We study the Galois tower generated by iterates of a quadratic polynomial $f$ defined over an arbitrary field. One question of interest is to find the proportion $a_n$ of elements at level $n$ thatExpand
The Density of Primes in Orbits of zd+c
Given a polynomial f(z) = z^d + c over a global field K and a_0 in K, we study the density of prime ideals of K dividing at least one element of the orbit of a_0 under f. The density of such sets forExpand
Fixed-point-free elements of iterated monodromy groups
The iterated monodromy group of a post-critically finite complex polynomial of degree d \geq 2 acts naturally on the complete d-ary rooted tree T of preimages of a generic point. This group, as wellExpand
Profinite iterated monodromy groups arising from quadratic morphisms with infinite postcritical orbits
We study in detail the profinite group G arising as geometric \'etale iterated monodromy group of an arbitrary quadratic morphism f with an infinite postcritical orbit over a field of characteristicExpand
Profinite iterated monodromy groups arising from quadratic polynomials
We study in detail the profinite group G arising as geometric \'etale iterated monodromy group of an arbitrary quadratic polynomial over a field of characteristic different from two. This is aExpand
Eventually stable rational functions
For a field K, rational function phi in K(z) of degree at least two, and alpha in P^1(K), we study the polynomials in K[z] whose roots are given by the solutions to phi^n(z) = alpha, where phi^nExpand
Finitely ramified iterated extensions
Let K be a number field, t a parameter, F = K(t), and '(x) ∈ K(x) a polyno- mial of degree d ≥ 2. The polynomialn(x, t) = ' ◦n (x)−t ∈ F(x), where ' ◦n = '◦'◦� � �◦ ' is the n-fold iterate of ', isExpand
Arboreal Galois representations and uniformization of polynomial dynamics
Given a polynomial f of degree d defined over a complete local field, we construct a biholomorphic change of variables defined in a neighbourhood of infinity which transforms the action z->f(z) toExpand
Primitive Divisors, Dynamical Zsigmondy Sets, and Vojta's Conjecture
A primitive prime divisor of an element a_n of a sequence (a_1,a_2,a_3,...) is a prime P that divides a_n, but does not divide a_m for all m X be a self-morphism of a variety, let D be an effectiveExpand
...
1
2
3
4
5
...