Arboreal Galois Representations and Applications to Arithmetic Dynamics

这个项目涉及到整体域的Galois扩张群的研究，这些扩张是由一个有理点在簇的态射迭代下的邻接前像得到的。所有这样的群对给定点和态射的逆极限形成了我们所说的树形伽罗瓦表示。关于这些表示的零碎结果已经存在了大约20年，但PI最近发展了一个更统一的理论，并发现了几个新的应用。其中包括p-进Mandelbrot集的性质，非线性递归的素因子集，以及交换代数群上给定点的约化。PI计划进一步应用于有限域上的动力学领域，这是一个自然的方向。包括Pollard Rho算法在内的几种密码算法利用了有限域上的动力学，这里提出的研究很可能在这一领域有应用。进一步的研究计划包括确定某些类似于CM椭圆曲线的树形表示的映象；进一步发展树形表示和线性伽罗瓦表示之间的类比，最终目的是将有趣的L函数附加到树形表示上；最后，检查整系数多项式迭代的不可约性。总的来说，这个项目融合了数论和动力学这两个先验不同领域的思想。研究有理数Q的代数扩张--即多项式的根--是数论最基本的领域之一。动力学领域试图了解过程是如何随时间演变的，而最基本的动力系统包括重复应用(或迭代，如所知的)从一个空间到它本身的映射f。PI建议研究由某些多项式的迭代的邻接根所得到的Q的扩张。特别令人感兴趣的是这类场的伽罗瓦群，即逐点固定基场的场自同构群。当一个函数的所有迭代的伽罗瓦群加在一起时，我们称它为树上伽罗瓦表示。即使在看似简单的情况下，如多项式x^2-1，这种树表示法也不能很好地理解。事实证明，这些表示法编码了关于各种动力学现象的密度信息。此外，它们提供了一个有趣的和潜在富有成效的类似于研究得很好的线性L-ADDY伽罗瓦表示的情况，即研究其伽罗瓦群嵌入某些矩阵群的域。这些已经有了无数重要的应用。

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'sknown) 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.