The density of prime divisors in the arithmetic dynamics of quadratic polynomials

We consider integer recurrences of the form a_n = f(a_{n-1}), where f is a quadratic polynomial with integer coefficients. We show, for four infinite families of f, that the set of primes dividing at least one term of such a sequence must have density zero, regardless of choice of a_0. The proof relies on tools from group theory and probability theory to develop a zero-density criterion in terms of arithmetic properties of the forward orbit of the critical point of f. This provides an analogy to results in real and complex dynamics, where analytic properties of the forward orbit of the critical point determine many global dynamical properties of f. The article also includes apparently new work on the irreducibility of iterates of quadratic polynomials.Comment: 21 pages, LaTe

An iterative construction of irreducible polynomials reducible modulo every prime

We give a method of constructing polynomials of arbitrarily large degree irreducible over a global field F but reducible modulo every prime of F. The method consists of finding quadratic f in F[x] whose iterates have the desired property, and it depends on new criteria ensuring all iterates of f are irreducible. In particular when F is a number field in which the ideal (2) is not a square, we construct infinitely many families of quadratic f such that every iterate f^n is irreducible over F, but f^n is reducible modulo all primes of F for n at least 2. We also give an example for each n of a quadratic f with integer coefficients whose iterates are all irreducible over the rationals, whose (n-1)st iterate is irreducible modulo some primes, and whose nth iterate is reducible modulo all primes. From the perspective of Galois theory, this suggests that a well-known rigidity phenomenon for linear Galois representations does not exist for Galois representations obtained by polynomial iteration. Finally, we study the number of primes P for which a given quadratic f defined over a global field has f^n irreducible modulo P for all n.Comment: 19 pages. Section 4 contains the construction of the examples mentioned in the abstract. Version 4 includes final revisions to the manuscrip

Galois representations from pre-image trees: an arboreal survey

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 size and structure of this Galois representation. These inquiries arose in part because knowledge of G(phi) allows one to prove density results on the set of primes of K that divide at least one element of a given orbit of phi. Following an overview of the history of the subject and two of its fundamental questions, we survey cases where G(phi) is known to have finite index in Aut(T). While it is tempting to conjecture that such behavior should hold in general, we exhibit four classes of rational functions where it does not, illustrating the difficulties in formulating the proper conjecture. Fortunately, one can achieve the aforementioned density results with comparatively little information about G(phi), thanks in part to a surprising application of probability theory. Underlying all of this analysis are results on the factorization into irreducibles of the numerators of iterates of phi, which we survey briefly. We find that for each of these matters, the arithmetic of the forward orbits of the critical points of phi proves decisive, just as the topology of these orbits is decisive in complex dynamics

Harmonic forms on manifolds with edges

Let $(X,g)$ be a compact Riemannian stratified space with simple edge singularity. Thus a neighbourhood of the singular stratum is a bundle of truncated cones over a lower dimensional compact smooth manifold. We calculate the various polynomially weighted de Rham cohomology spaces of $X$, as well as the associated spaces of harmonic forms. In the unweighted case, this is closely related to recent work of Cheeger and Dai \cite{CD}. Because the metric $g$ is incomplete, this requires a consideration of the various choices of ideal boundary conditions at the singular set. We also calculate the space of $L^2$ harmonic forms for any complete edge metric on the regular part of $X$