Special Values of L-Functions

摘要德克萨斯大学的fernando rodriguez - villegas教授将继续他关于l函数的特殊值的研究。他将探索最近发现的多项式的马勒测度与l函数的特殊值之间的联系。马勒测度出现在许多引人入胜且看似无关的背景中，如Arakelov理论中的变异高度、超越数论、代数动力系统的熵和离散算子的谱。在这个项目中，罗德里格斯-维勒加斯教授将尝试用我们对多项式的马勒测度的理解和这种与l函数的特殊值的新关系来阐明Bolch和Beilinson的猜想。这是现代数论的一个课题。对自然数基本性质的研究长期以来一直是数学的一部分。我们对这些基本性质不断增长的理解，已经导致了熟悉的数字系统的巧妙的实践和理论应用，远远超出了数学。数字的许多基本属性可以通过计算某些重要对象的集合来总结为数字本身。不幸的是，很多时候这些集合很难计数。数学家们已经发现，许多重要的计数可以通过将信息编码到可以通过微积分构造的l函数中来找到。这些l函数的一些特殊值给出了它们所寻找的答案。通过研究这些特殊值，罗德里格斯-维勒加斯教授希望建立对这些数字的估计，从而揭示它们总结的重要性质。

AbstractFernando Rodriquez-VillegasUniversity of TexasDMS-9970109Professor Rodriquez-Villegas will continue his study of the special values of L-functions. He will explore the recently discovered connection between the Mahler measure of polynomials and special values of L-functions. Mahler measure appears in many fascinating and seemingly unrelated contexts like heights of varieties in Arakelov theory, transcendental number theory, entropy of algebraic dynamical systems, and spectrum of discrete operators. In this project Professor Rodriquez-Villegas will try to use our understanding of Mahler measure of polynomials and this new relation to special values of L-functions to shed light on the conjectures of Bolch and Beilinson.This is a project in modern Number Theory. The study of the basic properties of the natural numbers has long been a part of mathematics. Our ever-increasing understanding of these basic properties has led to ingenious practical and theoretical applications of the familiar number systems well beyond mathematics. Many of the basic properties of numbers can be summarized into number themselves by counting certain collections of important objects. Unfortunately many times these collections are much too difficult to actually count. Mathematicians have discovered that many of the important counts can be found by encoding information into certain L-functions that can be constructed via calculus. Some special values of these L-functions give the answers they are looking for. By studying these special values, Professor Rodriquez-Villegas hopes to establish estimates of the numbers that will shed light on the important properties they summarize.