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

自同构形式理论是一个广泛而深入的学科，涉及数学的许多领域，如数论、调和分析和几何。朗兰兹的纲领是一个雄心勃勃的计划，以系统的方式发展自同构形式的理论。它开辟了数学的新领域，并为解决老问题提供了新的见解和技术。事实上，安德鲁·怀尔斯对费马大定理的解就是朗兰兹纲领的成就之一。近年来，该理论有了许多令人兴奋的新进展：L. Lafforgue利用Drinfeld的思想，在函数场上得到了GL(n)的全局朗兰兹对应；M. Harris和R. Taylor用Shimura变量和G. Henniart用l函数证明了p进域上GL(n)的局部Langlands对应；利用Langlands-Shahidi方法和Cogdell-Piatetski-Shapiro的逆定理，研究了H. Kim和F. Shahidi对GL(2)的尖形对称立方和H. Kim对GL(2)的尖形对称四次的Langlands泛函性。这是一个理想的时间有一个特殊的程序在自同态形式来回顾这些新的发展。它们在经典数论中有着深远的应用。特别是对称立方和对称四次方的朗兰兹泛函在解析数论中有直接的应用。事实上，这个项目的目标之一就是邀请自同构形式和解析数论的专家来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。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“解析的”。令人惊奇的是，正确类型的解析函数常常为离散问题提供答案。l函数是一种生成函数，由与数论相关的几何对象或一种“解析”函数（称为自同构形式）相关的数据组成，这些数据由线性变换群决定。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成为数论的一个分支。近年来，许多人发现了l函数编码离散信息的新方法，尽管仍有许多有待发现。这个项目的主要目的是将解析数论、自同构形式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.