The Relative Trace Formula and its Applications

相对微量公式及其应用

• 批准号: 0070779
• 负责人: Jonathan Rogawski
• 金额: $ 16.5万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070779&HistoricalAwards=false
• 关键词: Relative; Trace; Formula; its; Applications

ABSTRACTTECHNICAL DESCRIPTIONThe relative trace formula (RTF) is used to study automorphic representations of a reductive group G that are distinguished by a subgroup H obtained as the fixed-point set of an involution. One hopes to characterize distinguished generic representations as functorial transfers from a third group G' (which can be specified conjecturally in terms of the involution) by comparing RTF with the Kuznetzov trace formula on G'. To this end, it is important to obtain a fine spectral expansion of the RTF and, in particular, to write the spectral expansion in terms of relative Bessel distributions. Progress towards this goal was made in prior joint work by the PI, Jacquet and Lapid for the case of the standard Galois involution on GL(n). They developed a procedure for defining regularized periods of Eisenstein series, which turns out to be key ingredients in the fine spectral expansion. In some cases, it has been possible to express the regularized periods in terms of certain integrals analagous to intertwining operators which have been called intertwining periods. The PI intends, in collaboration with E. Lapid, to extend this previous work to the general case. This will involve analytic problems related to Eisenstein series and combinatorial problems related to the structure of the orbits of the Borel subgroup on G/H. It will also be necessary to develop a suitable truncation operator. One would like to express the regularized periods of cuspidal Eisenstein series in terms of L-functions. In general, however, the regularized period will be equal to an infinite sum of intertwining periods. To deal with this problem, the PI and Lapid intend to develop a formalism for forming linear combinations of the intertwining periods, in analogy with the linear combinations of characters that occur in the endoscopic theory of Langlands-Shelstad. Closely related is the problem of establishing identities between relative Bessel distributions for the pair (G,H) and Bessel distributions on G'. In a related project to be carried out with D. Ramakrishnan, the PI will investigate certain limit formulas connected with relative trace formulas. This will lead to a new method of proof and more precise versions of previously known results of W. Duke and others on the distribution of certain special values of GL(2) L-functions. The distribution results will involve certain measures on the spherical dual. Higher rank cases will be investigated and a general context in which to place the results will be sought.NON-TECHNICAL DESCRIPTIONThe history of mathematics has shown that the simplest phenomena are sometimes the hardest to understand deeply. The correct explanation may emerge only after the right theoretical framework has been found. The reciprocity laws of number theory fall into this category of mathematical phenomena. The simplest law of this type, the so-called law of quadratic reciprocity, is a beautiful and mysterious fact about ordinary whole numbers. It can be explained to a curious high school student, but its true structural meaning can only be understood within the context of a sophisticated and advanced part of number theory called class field theory. One of the great challenges of modern number theory is to fully explore the most general reciprocity laws. A framework for formulating such laws was developed 30 years ago by R. Langlands, and as a result, we know that there must exist a vast web of interrelated reciprocity laws. As a totality, these conjectural laws are called the functoriality principle. The functoriality principle seeks to explain the reciprocity laws within the context of a theory that originated in theoretical physics, the so-called representation theory of semisimple groups. In addition to ties with advanced theoretical physics, the theory of functoriality has found applications in diverse areas of combinatorics, coding theory, and cryptography. Enormous progess in the theory of functoriality has been made during the last thirty years which in turn has motivated much outstanding research, including the solution of the famous Fermat's Last Theorem. Despite this, our understanding of functoriality remains rudimentary in many respects. When a fully developed theory of functoriality is eventually developed, we can expect it to have a profound influence on mathematics and some areas of its applications. The goal of the project supported by this grant is to advance our understanding of the Relative Trace Formula, which is one of a handful of valuable tools that we have for studying functoriality. The results of this study will make it possible to study the functoriality principle from the point of view of "period integrals". Hopefully, this will play a role in advancing our knowledge of the general functoriality principle.

摘要技术描述：利用相对迹公式（RTF）研究了约化群G的自同构表示，这些约化群由一子群H作为对合的不动点集来区分。人们希望通过比较RTF和G‘上的库兹涅佐夫迹公式，将不同的一般表示表征为来自第三群G’（可以根据对合来推测地指定）的泛函转移。为此，重要的是获得RTF的精细光谱展开，特别是用相对贝塞尔分布来表示光谱展开。在此之前，PI、Jacquet和Lapid就GL(n)上的标准伽罗瓦对合进行了联合研究，在这方面取得了进展。他们开发了一个定义爱森斯坦级数正则周期的程序，这被证明是精细光谱展开的关键成分。在某些情况下，可以用某些积分来表示正则周期，类似于被称为交织周期的缠结算子。PI打算与E. Lapid合作，将以前的工作扩展到一般情况。这将涉及到与爱森斯坦级数有关的解析问题和与G/H上Borel子群轨道结构有关的组合问题。开发一个合适的截断运算符也是必要的。有人想用l函数来表示倒轴爱森斯坦级数的正则周期。然而，一般来说，正则化周期将等于纠缠周期的无限和。为了解决这个问题，PI和Lapid打算开发一种形式，用于形成相互交织的周期的线性组合，类似于Langlands-Shelstad的内窥镜理论中出现的字符的线性组合。与此密切相关的是建立（G，H）对的相对贝塞尔分布与G'上的贝塞尔分布之间的恒等式的问题。在与D. Ramakrishnan进行的一个相关项目中，PI将研究与相对迹公式相关的某些极限公式。这将导致一种新的证明方法和W. Duke等人关于GL(2) l函数的某些特殊值的分布的已知结果的更精确的版本。分布结果将涉及球面对偶上的某些度量。将对级别较高的案件进行调查，并寻求将结果置于其中的一般背景。数学的历史表明，最简单的现象有时是最难深入理解的。正确的解释可能只有在找到正确的理论框架之后才会出现。数论中的互易定律就属于这一类数学现象。这种类型中最简单的定律，即所谓的二次互易性定律，是关于普通整数的一个美丽而神秘的事实。它可以解释给一个好奇的高中生，但其真正的结构意义只能在数论的一个复杂和先进的部分被称为类场论的背景下理解。现代数论面临的巨大挑战之一是充分探索最普遍的互易律。朗兰兹（R. Langlands）在30年前制定了一个框架来表述这些定律，因此我们知道，一定存在着一个相互关联的互惠定律的庞大网络。作为一个整体，这些推测规律被称为功能原理。泛函原理试图在理论物理学的背景下解释互易定律，即所谓的半单群表示理论。除了与高级理论物理的联系外，功能理论在组合学、编码理论和密码学的各个领域都有应用。泛函理论在过去的三十年里取得了巨大的进步，这反过来又激发了许多杰出的研究，包括著名的费马大定理的解。尽管如此，我们对功能的理解在许多方面仍处于初级阶段。当一个完整的功能理论最终发展起来时，我们可以预期它会对数学及其应用的某些领域产生深远的影响。该基金支持的项目的目标是促进我们对相对跟踪公式的理解，相对跟踪公式是我们用于研究功能的少数有价值的工具之一。本研究结果将使从“周期积分”的角度研究泛函原理成为可能。希望这将有助于提高我们对一般功能原理的认识。