Arboreal Galois Representations and Applications to Arithmetic Dynamics

College Of The Holy Cross, Worcester MA

Investigators

Abstract

This project involves the study of Galois groups of extensions of global fields obtained by adjoining preimages of a rational point under iterates of a morphism of varieties. The inverse limit of all such groups for a given point and morphism form what we call an arboreal Galois representation. Piecemeal results on these representations have existed for some 20 years, but the PI has recently developed a more unified theory, and discovered several new applications. These include properties of the p-adic Mandelbrot set, sets of prime divisors of non-linear recurrences, and reductions of a given point on an abelian algebraic group. The PI plans to pursue applications further, into the domain of dynamics over finite fields, which is a natural direction. Several cryptographic algorithms, including the Pollard rho algorithm, make use of dynamics over finite fields, and the research proposed here is likely to have applications in this area. Further research plans include the determination of the image of arboreal representations in certain analogues to the case of CM elliptic curves; the further development of the analogy between arboreal representations and linear Galois representations, with the ultimate goal of attaching interesting L-functions to arboreal representations; and finally, an examination of irreducibility properties of iterates of polynomials with integral coefficients. Generally speaking, this projects blends ideas from two a priori different fields, number theory and dynamics. The study of extensions of the rational numbers Q by algebraic numbers -- that is, roots of polynomials -- is one of the most basic areas of number theory. The field of dynamics seeks to understand how processes evolve over time, and the most basic dynamical system consists of repeated application (or iteration, as it's known) of a map f from a space to itself. The PI proposes to study the extensions of Q obtained by adjoining roots of iterates of certain polynomials. Of particular interest are the Galois groups of such fields, namely the group of field automorphisms fixing pointwise the base field. When the Galois groups of all iterates of a single function are taken together, we term it an arboreal Galois representation. Even in seemingly simple cases such as the polynomial x^2 - 1, this arboreal representation is not well understood. These representations turn out to encode density information regarding a variety of dynamical phenomena. Moreover, they furnish an interesting and potentially fruitful analogue to the well-studied case of linear l-adic Galois representations, namely the study of fields whose Galois groups embed in certain matrix groups. These have had a myriad of important applications.