Arboreal Galois Representations and Applications to Arithmetic Dynamics

这个项目涉及到研究伽罗瓦群的扩展的全球领域所获得的邻接原像的一个合理的点下迭代态射的品种。 所有这样的群对于一个给定点的逆极限和态射形成了我们所说的树实伽罗瓦表示。 关于这些表示的零碎结果已经存在了大约20年，但PI最近开发了一个更统一的理论，并发现了一些新的应用。这些包括属性的p进曼德尔布罗特集，集的主要因子的非线性递归，并减少了一个给定的点上的阿贝尔代数群。PI计划进一步追求应用，进入有限域上的动力学领域，这是一个自然的方向。几个密码算法，包括波拉德rho算法，利用有限域上的动态，这里提出的研究很可能在这一领域的应用。进一步的研究计划包括确定树表示在某些类似CM椭圆曲线的情况下的图像;树表示和线性伽罗瓦表示之间的类比的进一步发展，最终目标是将有趣的L-函数附加到树表示;最后，研究整系数多项式迭代的不可约性。一般来说，这个项目融合了两个先验不同领域的思想，数论和动力学。 研究有理数Q的代数数扩张--即多项式的根--是数论最基本的领域之一。 动力学领域试图理解过程如何随时间演化，最基本的动力学系统由从空间到自身的映射f的重复应用（或迭代）组成。 PI建议研究通过某些多项式的迭代的邻接根获得的Q的扩展。 特别令人感兴趣的是伽罗瓦集团等领域，即集团的领域自同构固定点的基本领域。当一个函数的所有迭代的伽罗瓦群放在一起时，我们称之为树型伽罗瓦表示。 即使在看似简单的情况下，如多项式x^2 - 1，这种树坐标表示也不是很好理解。 这些表示原来编码密度信息有关的各种动力学现象。 此外，他们提供了一个有趣的和潜在的富有成效的类似研究的情况下，线性l-进伽罗瓦表示，即研究领域的伽罗瓦群嵌入在某些矩阵群。它们有无数重要的应用。

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.