Stark--type Conjectures "over Z" and their Refined Versions

christian Popescu德克萨斯大学Austin 98 01267 Popescu教授将研究Stark的l函数猜想与欧拉系统构造之间的联系。研究者将研究关于高阶消失的猜想的泛函行为，寻找这些猜想的精炼版本，并考虑它们与欧拉系统的联系。本世纪下半叶最重要的数学思想之一，是解析公式经常编码离散信息。例如，一个人可能想要计算一个特定方程的解的个数，但是发现解很难找到。另一方面，您可能想要计算一系列方程的解的数量，并将答案作为一个序列。数学家称这类问题为“离散问题”。微积分中出现的函数，并不是“离散的”，而是“解析的”。令人惊奇的是，正确的解析函数可以包含离散例子的答案或答案。这个项目涉及到l函数，它是这类解析函数的第一个例子。在过去的30年里，包括H. Stark在内的许多人发现了l函数编码离散信息的新方法。因为这些联系中的许多仍然只是猜想，所以l函数在当今数学中非常重要。

ABSTRACT Cristian Popescu University of Texas at Austin 98 01267 Professor Popescu will study connections between Stark's conjecture for L-functions and the construction of Euler systems. The investigator will study the functorial behavior of the conjectures concerning higher order of vanishing, looking for refined versions of these conjectures, and consider their connections with Euler systems. One of the most important mathematical ideas of the second half of the century, is that analytic formula often encodes discrete information. For example, one might want to count the number of solutions of a particular equation, but discover that the solutions are very hard to find. On the other hand, you might want to count the number of solutions to a sequence of equations and have the answers as a sequence. Mathematicians call these kind of problems 'discrete'. The functions that appear in calculus, and for which calculus works so well, are not 'discrete', but 'analytic.' Amazingly, the right kind of analytic function can contain the answer or answers to the discrete examples about. This project involves L-functions which were among the first examples of this kind of analytic function. In the past thirty years, a number of people including H. Stark have found new ways L-functions encode discrete information. Because many of these connections are still only conjecture, L-functions are very important in current mathematics.