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

在这个项目中，Dinakar Ramakrishnan教授将研究数论中Langlands程序的三个核心技术问题。一个问题涉及到泛函原理，涉及到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一个猜想。最后的专题是了解一类具有正无穷型的复乘法域上的尖形。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。数学家称这类问题为“离散”，因为你在计算分离的事物。微积分中出现的函数，以及微积分非常擅长的函数，都不是离散的，而是解析的，因为它们处理的是所有实数，不管它们之间的距离有多近。令人惊奇的是，正确的解析函数可以包含一个离散例子的答案。实际上第一个这样的例子是在几百年前发现的，但是在过去的三十年里，这些例子被系统化成为数论的一个分支，叫做朗兰兹程序。

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.