Anabelian Geometry and Elementary Equivalence of Fields

波普DMS-0401056奖摘要本研究项目涉及现代伽罗瓦理论，更具体地说，是二元亚阿贝尔几何。我将要研究的主要问题是试图从L关于它的函数域的伽罗瓦理论中恢复簇的二元类。与此相关的还有其他几个问题，比如有理数域上的伽罗瓦群的几何/组合描述。上述问题也与通过所谓的截面猜想来描述簇的有理点有关。这类问题是在算术情境中由Grothendieck在可自由标记(未发表)的手稿中提出的。但是似乎在更高的维度上，一个人即使在完全没有算术的情况下也有这样的“再生现象”，也就是在代数封闭的基域上。伽罗瓦理论在两个非常不同的数学方面之间架起了一座桥梁，即一边是一些基本的代数对象，如场，或者更一般的空间(簇)，另一边是求解代数方程或所讨论空间的构造覆盖的方法。现在，(两国的)阿贝尔几何断言，在田野工作的情况下是“足够原始的”，分别一个构造所覆盖的空间的几何是“足够复杂的”，分别构造所有覆盖的所有代数方程的解的“配方”的层次性分别编码了场，分别讨论的空间。这在探讨一些非常基本的数学问题方面打开了一个全新的视角。本研究项目解决了这一数学研究领域的一些基本问题。

ABSTRACT for Award DMS-0401056 of PopThe present research project concerns modern Galois Theory,more specifically birational Anabelian Geometry. The mainproblem I will be investigating is trying to recover thebirational class of a variety (over an algebraically closedbase field) from the pro-l Galois theory of its function field.Related to this, there are several other questions, like theone by Ihara/Oda-Matsumoto concerning a geometric/combinatoricdescription of the Galois group of the field of rational numbers.The above question is also related to describing rational pointsof varieties via the so called Section Conjecture. This kind ofquestions were initiated -in the arithmetic situation- in someremarkable (unpublished) manuscripts by Grothendieck. But itappears that in higher dimensions, one has such "anabelianphenomena" even in the total absence of arithmetic, i.e., overan algebraically closed base field.The Galois Theory makes a bridge between two very differentmathematical aspects, namely some basic algebraic objects, likefields, or more general spaces (varieties) on the one side,and the way one solves algebraic equations, or constructscovers of the spaces in discussion, on the other side. Now the(birational) Anabelian Geometry asserts that in the case the fieldone works over is "primitive enough", respectively the geometryof the space one constructs covers of, is "complicated enough", thetotality of the "recipes" of solving all the algebraic equations,respectively of constructing all the covers, encodes the field,respectively the space under discussion. This opens a completelynew perspective in approaching some very fundamental mathematicalquestions. The present research project addresses some of the basicproblems in this mathematical field of research.