Asai L-Functions, Forms on GL(4), and Applications

Asai L 函数、GL(4) 形式和应用

• 批准号: 9801328
• 负责人: Dinakar Ramakrishnan
• 金额: $ 16万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2002-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9801328&HistoricalAwards=false
• 关键词: Asai; Functions; Forms; GL; Applications

ABSTRACT Dinakar Ramakrishnan Cal Tech DMS 98-01328 In this project, Professor Dinakar Ramakrishnan will study three technical problems central to the Langlands program in Number Theory. One problem involves the principle of functoriality involving the Asai L-functions and automorphic forms on the general linear group of degree 4. The second involves a conjecture of the famous mathematician from the early part of the century, Ramanujan. The final project is to understand a class of cusp forms over complex multiplication fields with regular infinity type. 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. Mathematicians call these kind of problems 'discrete' because you are counting separated things. The functions that appear in calculus, and for which calculus works so well, are not 'discrete', but 'analytic' because they deal with all the real numbers no matter how close together. Amazingly, the right kind of analytic function can contain the answer a discrete examples. Actually the first examples of this were discovered a couple hundred years ago, but in the past thirty years, these examples have been systematized into a branch of number theory called the Langlands program.

摘要 Dinakar Ramakrishnan 加州理工学院 DMS 98-01328 在该项目中，Dinakar Ramakrishnan 教授将研究数论中朗兰兹计划的三个核心技术问题。 其中一个问题涉及函子性原理，涉及 4 次一般线性群上的 Asai L 函数和自同构形式。第二个问题涉及本世纪初著名数学家拉马努金的猜想。最终项目是理解具有规则无穷类型的复杂乘法域上的一类尖点形式。 本世纪下半叶最重要的数学思想之一是解析公式通常编码离散信息。例如，人们可能想要计算某个特定方程的解的数量，但发现解很难找到。数学家将此类问题称为“离散”问题，因为您正在计算分离的事物。微积分中出现的函数以及微积分非常有效的函数不是“离散”函数，而是“解析”函数，因为它们处理所有实数，无论它们之间的距离有多近。 令人惊讶的是，正确类型的分析函数可以包含离散示例的答案。 实际上，第一个例子是在几百年前发现的，但在过去的三十年里，这些例子已经被系统化为数论的一个分支，称为朗兰兹纲领。