The Relative Trace Formula and its Applications

利用相对迹公式（RTF）研究了约化群G的自守表示，它以子群H作为对合的不动点集。 人们希望通过比较RTF与G '上的Kuznetzov迹公式，将区别的一般表示表征为来自第三个群G'（可以用对合来描述）的函子转移。 为此，重要的是获得RTF的精细谱展开，特别是，以相对贝塞尔分布的形式写出谱展开。PI、Jacquet和Lapid在GL（n）上的标准伽罗瓦对合的情况下，在之前的联合工作中取得了进展。 他们开发了一个程序来定义爱森斯坦级数的正则化周期，这被证明是精细谱展开的关键成分。在某些情况下，正则化周期可以用某些积分来表示，这些积分类似于称为交织周期的交织算子。 PI打算与E. Lapid，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分析问题和与G/H上Borel子群轨道结构有关的组合问题。还需要开发合适的截断运算符。 人们希望用L-函数来表示尖点艾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总和。 为了解决这个问题，PI和拉皮德打算发展一种形式主义，以形成交织周期的线性组合，类似于Langlands-Shelstad的内窥镜理论中出现的字符的线性组合。与此密切相关的问题是建立对（G，H）的相对贝塞尔分布与G '上的贝塞尔分布之间的恒等式。 在与D. Ramakrishnan，PI将研究与相关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更精确的版本以前已知的结果W。杜克等人关于GL（2）L-函数某些特殊值的分布。 分布结果将涉及到球对偶上的某些测度。更高级别的情况下将进行调查，并在其中放置的结果将寻求一个一般的上下文。非技术性的解释数学的历史表明，最简单的现象有时是最难理解的。 正确的解释只有在找到正确的理论框架之后才可能出现，数论的互反律就属于这一类数学现象。 这种类型的最简单的定律，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。 它可以解释给一个好奇的高中生，但它真正的结构意义只能在数论的一个复杂和高级部分的背景下理解，称为类域理论。 现代数论的一个重大挑战是充分探索最普遍的互反律。 30年前，R.因此，我们知道必然存在着一个由相互关联的互惠律组成的庞大网络。 作为一个整体，这些自然规律被称为功能性原则。 函性原理试图在理论物理学中的一个理论的背景下解释互反律，即所谓的半单群表示理论。 除了与高级理论物理学的联系外，函数性理论还在组合学、编码理论和密码学的各个领域中得到应用。 在过去的三十年里，函数性理论取得了巨大的成就，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方案。 尽管如此，我们对功能性的理解在许多方面仍然是初步的。 当一个充分发展的函数性理论最终发展起来时，我们可以期待它对数学及其应用的某些领域产生深远的影响。 该项目的目标是支持这笔赠款是推进我们对相对迹公式的理解，这是我们研究函性的少数宝贵工具之一。 本文的研究结果将为从“周期积分”的角度研究泛函原理提供可能。 希望，这将在推进我们对一般函数性原则的认识方面发挥作用。

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.