Special Semester Program on Automorphic Forms, Shimura Varieties and L-functions; January 1-May 31, 2003, Fields Institute, Toronto, Canada

自同构形理论是一个涉及数论、调和分析和几何等多个数学领域的广泛而深入的学科。朗兰兹的计划是一项雄心勃勃的计划，旨在系统地发展自同构形式理论。它开辟了数学的新前沿，并为解决旧问题提供了新的见解和技术。事实上，由Andrew Wiles提出的费马大定理的解是朗兰兹程序的成果之一.最近在这一理论中有许多令人兴奋的新发展：L.Lforgue利用Drinfeld的思想得到了函数域上GL(N)的全局朗兰兹对应；M.Harris和R.Taylor利用Shimura变数以及G.Henniart利用L函数得到了p-adady域上GL(N)的局部朗兰兹对应；用朗兰兹-沙希迪的方法和Cogdell-Piatetski-Shapiro的逆定理，得到GL(2)的尖形对称立方的朗兰兹函数和H.Kim的GL(2)的对称四次尖形的函数。这是一个理想的时机，有一个特别的计划，以自我同构的形式来审查这些新的发展。这些在经典数论中有着深远的应用。特别是朗兰兹的对称立方函数和对称四次函数在解析数论中有直接的应用。事实上，该计划的目标之一是将自同构形式和解析数论的专家聚集在一起，寻找自同构形式在解析数论中的应用，反之亦然。19世纪下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。另一方面，人们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“分析的”，对这些函数来说，微积分运行得很好。令人惊讶的是，正确的解析函数往往能为离散问题提供答案。L函数是一种生成函数，它由与数论有关的几何对象或由线性变换组确定的一种被称为自同构形式的“解析”函数的数据组成。这些L函数提供了正确类型的解析函数。在过去的三十年里，对L函数的这些例子的研究已经系统化，形成了数论的一个分支。近年来，许多人发现了L函数编码离散信息的新方法，但仍有许多有待发现的地方。这个程序的主要目的是将解析数论、自同构形和几何方面的专家聚集在一起，以便我们可以找到更多用L函数编码算术信息的方法。

The theory of automorphic forms is a wide and deep subject touching many areas of mathematics, such as number theory, harmonic analysis and geometry. Langlands' program is an ambitious plan to develop the theory of automorphic forms in a systematic way. It opened a new frontier in Mathematics, and has given new insights and techniques in solving old problems. In fact, the solution of Fermat's last theorem by Andrew Wiles is one of the achievements of the Langlands' program.There have been many exciting new developments in the theory recently: the global Langlands correspondence for GL(n) over a functiona field by L. Lafforgue, using the ideas of Drinfeld; the local Langlands correspondence for GL(n) over a p-adic field by M. Harris and R. Taylor, using Shimura varieties, and by G. Henniart, using L-functions; Langlands' functoriality for symmetric cube of cusp form of GL(2) by H. Kim and F. Shahidi, and symmetric fourth of cusp form of GL(2) by H. Kim, using The Langlands-Shahidi method and the converse theorem of Cogdell-Piatetski-Shapiro. It is an ideal time to have a special program in automorphic forms to review these new developments. These have far-reaching applications in classical number theory. Especially, Langlands' functoriality of symmetric cube and symmetric fourth have direct applications to analytic number theory. In fact, one of the goals of the program is to bring experts in automorphic forms and analytic number theory to find applications of automorphic forms in analytic number theory, vice versa. One of the most important mathematical ideas of the second half of the 19th century is that an 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, one might want to count the number of solutions to a sequence of equations and have the answers as a sequence. Mathematicians call these kinds of problems 'discrete'. The functions that appear in calculus, and for which calculus works so well, are not 'discrete', but 'analytic.' Amazingly, the right kinds of analytic functions often provide the answers to the discrete problems. An L-function is a type of generating function formed out of data associated with either a geometric object that is related to number theory or with a kind of 'analytic' function, called an automorphic form, which are determined by groups of linear transformations. These L-functions provide the right kinds of analytic functions. In the past thirty years, the study of these examples of L-functions has been systematized into a branch of number theory. In recent years many people have found new ways that L-functions encode discrete information, though there is still much to be discovered. The chief purpose of this program is to bring together experts in analytic number theory, automorphic forms, and geometry so that we may find more of the ways that arithmetic information is encoded by L-functions.