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

Dinakar Ramakrishnan加州理工学院DMS 98-01328在这个项目中，Dinakar Ramakrishnan教授将研究数论中朗兰兹计划的三个核心技术问题。其中一个问题涉及到4次一般线性群上的Asai L函数和自同构型的函数性原理。第二个问题涉及本世纪初著名数学家拉马努扬的一个猜想。最后一个项目是理解一类正则无穷类型复乘法域上的尖点形式。本世纪后半叶最重要的数学思想之一是，解析公式通常对离散信息进行编码。例如，一个人可能想要计算一个特定方程的解的数量，但发现很难找到解。数学家称这类问题为“离散的”，因为你是在计算分开的东西。微积分中出现的函数不是“离散的”，而是“解析的”，因为它们处理所有的实数，无论它们有多近。令人惊讶的是，正确的解析函数可以包含一个离散的例子的答案.实际上，最早的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，形成了数论的一个分支，称为朗兰兹程序。

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.