The Fekete-Szego Theorem on Curves, with Splitting Conditions

研究者和他的学生学习能力理论及其在算术几何中的应用。容量是集合大小的度量，它出现在势理论中，在概率论、复分析和数论中都有应用。本课题的主要目标是在代数曲线上证明Fekete-Szego定理的一个非常强的版本，该定理断言，如果曲线上的一个阿得利集具有足够大的容量，那么在阿得利意义上，在它附近存在所有共轭的代数点。新定理将是一个基本的存在性定理，它产生的代数点受实理性和p进理性条件以及拓扑约束的约束。为了确定概念，对于复平面上的集合，圆的容量等于它的半径，线段的容量是它长度的四分之一。在分析中，容量为0的集合和正容量的集合的主要区别是：容量为0的集合对于全纯函数是“不可见的”。在数论中，主要的区别是容量大于1和小于1的集合。Fekete和Szego的经典定理指出，对于复共轭稳定的复平面上的一个集合，如果该集合的容量大于1，则该集合的每一个邻域都包含无限多个代数整数的伽罗瓦轨道。David Cantor将该定理推广到射影线上的阿德利集，随后研究者将其推广到代数曲线上的阿德利集。作为应用，研究者证明了仿射代数变量上代数整数点的存在性定理，这是Moret-Bailly和Szpiro及其学生大量工作的对象。早先，Raphael Robinson在另一个方向上扩展了Fekete-Szego定理，证明了如果在实直线上包含一个大于1的容量集，那么每一个实邻域都包含无限多个完全实数代数整数的伽罗瓦轨道。最近，研究者证明了鲁宾逊定理的一个阿德利克版本，证明了在有限的位置上存在完全实数和完全p进的代数数。这个项目的目标是将这个定理推广到代数曲线。所使用的方法将包括p进分析、势理论和代数函数的近似理论。丢芬图方程（寻找多个变量多项式方程的整数解）的研究是一个非常古老和非常困难的课题，可以追溯到希腊。只是在过去的半个世纪里，利用现代数论的方法才取得了很大的进展。两个著名的结果是“希尔伯特第十问题”（1970年由Matiyasevich提出）的负解，该问题询问是否存在一种算法来确定给定方程是否有整数解；以及费马大定理（怀尔斯于1995年提出）的否定解，该定理问的是整数的n次幂和是否可以是n次幂。研究者的工作追求一个不同的方向，表明对于比整数大得多的算术域，在适当的条件下确实存在解；此外，还有算法可以判断它们是否存在。目前的工作将大大减少已知存在解决方案的领域的规模。

The investigator and his students study capacity theory and its applications to arithmetic geometry. Capacity is a measure of size for sets, which arises in potential theory and has applications in probability, complex analysis, and number theory. The chief goal of this project is to prove a very strong version of the Fekete-Szego theorem on algebraic curves, which asserts that if an adelic set on a curve has large enough capacity, then there exist algebraic points with all their conjugates near it, in an adelic sense. The new theorem will be a basic existence theorem producing algebraic points which are subject to real and p-adic rationality conditions, as well as topological constraints. To fix ideas, for sets in the complex plane, the capacity of a circle turns out to equal its radius, and the capacity of a line segment is a quarter of its length. In analysis, the primary distinction is between sets of capacity 0 and sets of positive capacity: sets of capacity 0 are 'invisible' to holomorphic functions. In number theory, the main distinction is between sets of capacity greater than 1, and less than 1. The classical theorem of Fekete and Szego says that for a set in the complex plane stable under complex conjugation, if the capacity of the set is greater than 1, then every neighborhood of the set contains infinitely many Galois orbits of algebraic integers. David Cantor generalized the theorem to adelic sets on the projective line, and subsequently the investigator generalized it to adelic sets on algebraic curves. As an application, the investigator proved an existence theorem for algebraic integer points on affine algebraic varieties, which has been the object of considerable work by Moret-Bailly and Szpiro and their students. Earlier, Raphael Robinson had extended the Fekete-Szego theorem in another direction, showing that if a set of capacity greater than 1 were contained in the real line, then every real neighborhood contained infinitely many Galois orbits of totally real algebraic integers. Recently the investigator proved an adelic version of Robinson's theorem, proving the existence of algebraic numbers which were totally real and totally p-adic at a finite number of places. The goal of the project is to generalize this theorem to algebraic curves. The methods used will involve p-adic analysis, potential theory, and approximation theory for algebraic functions. The study of diophantine equations (looking for integer solutions to polynomial equations in several variables) is a very old and very difficult subject, going back to the Greeks. It is only within the last half-century that much progress has been made, using methods of modern number theory. Two famous results were a negative solution to "Hilbert's Tenth problem" (by Matiyasevich in 1970), which asks if there is an algorithm for determining whether or not a given equation has integer solutions; and the negative resolution of Fermat's Last Theorem (by Wiles in 1995), which asks if sums of n-th powers of integers can be n-th powers. The investigator's work pursues a different direction, showing that for much larger arithmetic domains than the integers, under appropriate conditions there do exist solutions; and moreover there exist algorithms for telling whether or not they exist. The current work will considerably reduce the size of the domains where solutions are known to exist.