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

斯塔克型“Z之上”猜想及其改进版本

• 批准号: 9801267
• 负责人: Cristian Popescu
• 金额: $ 6.71万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-06-01 至 2001-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9801267&HistoricalAwards=false
• 关键词: Stark; type; Conjectures; over; their

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.

摘要 得克萨斯大学奥斯汀分校Cristian Popescu 98 01267 Popescu教授将研究L函数的斯塔克猜想和欧拉系统的构建之间的联系。 研究人员将研究函子的行为有关高阶消失，寻找这些函子的改进版本，并考虑它们与欧拉系统的连接。 世纪后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，一个人可能想计算一个特定方程的解的数量，但发现很难找到解。另一方面，你可能想计算一个方程序列的解的数量，并将答案作为一个序列。数学家称这类问题为“离散”问题。出现在微积分中的函数，以及微积分如此有效的函数，不是“离散的”，而是“分析的”。“令人惊讶的是，正确的解析函数可以包含离散例子的答案。 该项目涉及L-函数，这是这种解析函数的第一个例子。 在过去的三十年里，包括H.斯塔克发现了L函数编码离散信息的新方法。由于其中许多联系仍然只是猜测，因此L函数在当前数学中非常重要。