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.
The work funded by this grant has blended 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. Central objects in this study are Galois groups, which measure the algebraic independence of the roots of a polynomial. 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. Some of the most important dynamical objects are iterated preimages of a point P, namely those points eventually mapping to P under some number of iterations of f, and pre-periodic points, or those that eventually map into cycles under iteration of f. During the period of this grant, the PI has applied number-theoretic methods to questions arising from dynamics, studying in particular Galois groups arising from iterated preimages of points under a map given by a rational function. These are called arboreal Galois representations. Through the PI's individual efforts as well as collaborations with numerous other mathematicians, many theorems have been proved over the course of this grant, including: showing the arboreal Galois representation is as large as possible in certain circumstances (see the accompanying diagram, which shows the preimages of zero under the first two iterations of a rational function to which the result applies); giving a construction of polynomials whose iterates are irreducible, but become reducible when considered modulo prime number; showing that rational functions with the property that all their critical points are pre-periodic are quite rare, in that only finitely many dynamically distinct maps of this kind exist with coefficients in Q; and making headway towards linking the study of arboreal Galois representations over a finite field, which is a rather new field, to properties of iterated maps over the complex numbers, where there is a very well-developed and powerful theory. This last set of results, in particular, may eventually give insights that will be relevant to certain algorithms such as the Pollard rho algorithm, which play roles in cryptography. The PI has made it a priority to involve undergraduates in the work done for this grant. Two senior theses and a summer research project for a group of three students have resulted from this work. Both of the theses involved the students doing original research, and the summer project resulted in a manuscript, co-authored with me and another professor, that will be submitted for publication. The PI has enjoyed seeing the students become excited about mathematical research.
