24,945 research outputs found
Quotients of trees for arithmetic subgroups of PGLâ‚‚ over a rational function field
In this note we determine the structure of the quotient of the Bruhat-Tits tree of the locally compact group PGL(2)(F-p) with respect to the natural action of its S-arithmetic subgroup PGL(2)(O-{p}), where F is a rational function field over a finite field and p is a place of F
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
- …