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.

摘要技术描述相对踪公式（RTF）用于研究还原群 G 的自同构表示，该表示通过作为对合的不动点集获得的子群 H 来区分。 人们希望通过将 RTF 与 G' 上的 Kuznetzov 迹公式进行比较，将不同的泛型表示表征为来自第三组 G'（可以根据对合推测地指定）的函数转移。 为此，重要的是获得 RTF 的精细谱展开，特别是根据相对贝塞尔分布来写出谱展开。 PI、Jacquet 和 Lapid 之前针对 GL(n) 的标准伽罗瓦对合案例的联合工作在实现这一目标方面取得了进展。 他们开发了一种定义爱森斯坦级数正则化周期的程序，该程序被证明是精细光谱扩展的关键成分。在某些情况下，可以用类似于交织算子的某些积分来表达正则化周期，这些积分被称为交织周期。 PI 打算与 E. Lapid 合作，将之前的工作扩展到一般情况。 这将涉及与爱森斯坦级数相关的解析问题以及与G/H上的Borel子群轨道结构相关的组合问题。还需要开发合适的截断运算符。 人们想用 L 函数来表达尖爱森斯坦级数的正则化周期。 然而，一般来说，正则化周期将等于交织周期的无限和。 为了解决这个问题，PI 和 Lapid 打算开发一种形式主义，用于形成交织周期的线性组合，类似于 Langlands-Shelstad 内窥镜理论中出现的特征线性组合。密切相关的是在 (G,H) 对的相对贝塞尔分布与 G' 上的贝塞尔分布之间建立恒等式的问题。 在与 D. Ramakrishnan 合作开展的一个相关项目中，PI 将研究与相对迹公式相关的某些极限公式。这将带来一种新的证明方法，以及 W. Duke 和其他人关于 GL(2) L 函数的某些特殊值的分布的先前已知结果的更精确版本。 分布结果将涉及球对偶的某些测量。将调查更高级别的案例，并寻求放置结果的一般背景。非技术描述数学的历史表明，最简单的现象有时是最难深入理解的。 只有找到正确的理论框架之后，正确的解释才可能出现。 数论的互易定律属于此类数学现象。 这种类型中最简单的定律，即所谓的二次互反定律，是关于普通整数的一个美丽而神秘的事实。 它可以向一个好奇的高中生解释，但它真正的结构含义只能在数论中一个复杂而高级的部分（称为类域论）的背景下才能理解。 现代数论的巨大挑战之一是充分探索最普遍的互易定律。 R. Langlands 于 30 年前开发了制定此类法律的框架，因此，我们知道必定存在一个相互关联的互惠法律的庞大网络。 作为一个整体，这些猜想定律被称为函子性原理。 函子原理试图在起源于理论物理学的理论（即所谓的半单群表示论）的背景下解释互反律。 除了与先进理论物理学的联系之外，函子性理论还在组合学、编码理论和密码学的不同领域中找到了应用。 在过去的三十年里，函子性理论取得了巨大的进步，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决。 尽管如此，我们对函数性的理解在许多方面仍然处于初级阶段。 当一个完全发展的函子性理论最终发展出来时，我们可以预期它将对数学及其应用的某些领域产生深远的影响。 这笔赠款支持的项目的目标是增进我们对相对迹公式的理解，这是我们研究函子性的少数有价值的工具之一。 这项研究的结果将使从“周期积分”的角度研究函子性原理成为可能。 希望这将有助于增进我们对一般函数性原理的了解。