Automorphic Forms: L-Functions and Related Geometry

本次会议计划于2012年4月23日至27日在康涅狄格州纽黑文的耶鲁大学举行。它将总结，综合，并展望未来，皮亚特茨基-夏皮罗在自同构形式，特别是l -函数领域的工作。会议为期五天，每天有五场演讲。讲座将探讨反映piatetki - shapiro主要研究领域的5个主题：泛函性和逆定理，显式结构和周期，p进l函数，几何和解析数论。每个主题将贯穿整个会议。将编写一卷论文集，目的是使这些结果更容易获得和适用。更多的信息可以在会议网站上找到：http://www.math.yale.edu/automorphicforms2012Automorphic形式是数论众多分支中最迷人和最神秘的一个。尽管它们很复杂，但它们有着显著而广泛的应用。a .怀尔斯和R.泰勒最近用自同构形式理论的结果证明了“费马大定理”——整数的两个完全n次幂的和不可能是完全n次幂，如果幂n大于2——这个问题已经被解决了300多年。自同构形式和l函数与素数密切相关，素数是大于1的整数，除了1和自身之外没有其他因子。它们是整数乘法的基础。质数最近被决定性地用于“公钥加密”，这是互联网上安全交易的基础。素数在所有整数中出现非常不规则，它们的分布问题吸引了大量的研究。黎曼ζ函数是l函数的第一个例子，它提供了一条途径来精确理解质数在所有正整数中的分布。l函数还提供了一种表达控制多项式方程解的微妙“互易律”的方法。自同构形式也是一些最美丽的数学公式的来源，例如，雅可比公式将整数表示为四个完全平方的和的方法数。最后，自同构形式理论与物理学之间有着显著的联系。同样的数学结构是量子力学的基础（海森堡规范对易关系），也是构造自同构形式的最重要方法之一的基础。此外，近年来从理论物理和相关数学中发现的各种公式与理论或自同构形式有很强的联系。皮亚茨基-夏皮罗是自同构形式理论的世界领袖。他的贡献的核心是建立自同构形式和l函数之间的密切联系，特别是通过他的“逆定理”，给出了l函数族与自同构形式相关的详细条件。这次会议提供了一个机会来综合、传播和建立皮亚茨基-夏皮罗的工作，以增加我们对这些迷人思想的理解。

This conference is planned to take place 23 - 27 April, 2012, at Yale University in New Haven, CT. It will summarize, synthesize, and project into the future, the work of I.I. Piatetski-Shapiro in the area of automorphic forms, especially L-functions. The conference will be of five days duration, with five talks per day. The talks will investigate 5 main themes that reflect Piatetski-Shapiro's main areas of investigation: functoriality and converse theorems, explicit constructions and periods, p-adic L-functions, geometry, and analytic number theory. Each theme will continue throughout the conference. A volume of proceedings, intended to make these results more accessible and applicable, will be produced. Additional information can be found at the conference website: http://www.math.yale.edu/automorphicforms2012Automorphic forms are one of the most fascinating and mysterious of the many branches of number theory. Despite their complexity, they have had remarkable and widely varied applications. Results from the theory of automorphic forms were used in the recent proof by A. Wiles and R. Taylor, of "Fermat's Last Theorem" -- the statement that the sum of two perfect nth powers of whole numbers cannot be a perfect n-th power, if the power n is greater than 2 -- a problem that had been unsolved for over 300 years. Automorphic forms and L-functions are deeply related to prime numbers, which are whole numbers larger than one that have no factors except 1 and themselves. They are the building blocks for multiplication of whole numbers. Prime numbers have recently been used decisively in "public key cryptography", which is the basis of secure transactions on the internet. Prime numbers occur very irregularly among all whole numbers, and the subject of their distribution has attracted an immense amount of research. The Riemann zeta function, which offers a path to a refined understanding of the distribution of prime numbers among all positive integers, is the first example of an L-function. L-functions also provide a means of expressing subtle "reciprocity laws" that govern the solutions of polynomial equations. Automorphic forms are also the source of some of the most beautiful formulas in mathematics, for example, Jacobi's formula for the number of ways to express a whole number as a sum of four perfect squares. Finally, there are remarkable connections between the theory of automorphic forms and physics. The same mathematical structure that is foundational to quantum mechanics (the Heisenberg Canonical Commutation Relations), is also the setting for one of the most important methods for constructing automorphic forms. In addition, a variety of formulas from theoretical physics and related mathematics have been discovered in recent years to have strong connections to the theory or automorphic forms. I. I. Piatetski-Shapiro was a world leader in the theory of automorphic forms. The heart of his contributions involved establishing close connections between automorphic forms and L-functions, especially through his "Converse Theorem", which gave detailed conditions for a family of L-functions to be related to an automorphic form. This conference offers an opportunity to synthesize, disseminate, and build on Piatetski-Shapiro's work, to increase our understanding of these fascinating ideas.