Arithmetic aspects of automorphic forms: Petersson norms and special values of L-functions

在过去的40年里，朗兰兹计划一直是数学领域的一个活跃研究领域，它由一个巨大的定理和猜想网络组成，这些网络连接着数论、表示论、分析和几何中的对象。朗兰兹计划的核心是自同构形式及其相关的L函数。这些对象产生于分析和表征理论。尽管取得了缓慢而稳定的进展，但关于自同构形和数论/算术几何之间的关系的许多基本问题仍然没有得到回答。这个项目的目的是通过对三个关键问题的深入研究来获得对其中一些问题的新见解：a)近全纯Siegel模形式，b)Deligne关于临界L值的代数性的猜想，c)函数相关自同构形式的Petersson范数之比。早期全纯Siegel模形式是由Shimura首先引入的，并且对于研究自同构L-函数的特殊值是不可或缺的。然而，尽管它们无处不在，但它们的算术性质和在朗兰兹框架中的位置还没有完全被理解。这个项目将研究它们的表示理论和算术性质，并证明它们与向量值Siegel模形式之间的密切联系。从近全纯模形式的研究中获得的见解将被用来解决由菲尔兹奖获得者数学家Pierre Deligne提出的一个著名猜想的一些特殊情况。Deligne猜想最简单的例子是一个经典的事实，即Riemann Zeta函数在所有正偶数时的值是pi乘以有理数的幂。这项研究将证明更复杂的L函数的预期合理性结果，这些函数在几个数学领域中具有重要意义。这项研究还旨在证明Ibukiyama和Katsurada关于某些Siegel模形式的Petersson范数之比的一个著名猜想。这个问题具有深远的意义，因为它涉及自同构形式在朗兰兹提升下的算术性质的行为。解决这个问题将需要对现有方法的创新适应。这个项目中使用的方法将是经典技术、现代表示理论方法和群上同调的强大组合。这项研究将为进一步探索开辟几条新的途径。

The Langlands program has been an area of active research in mathematics for the last forty years and consists of a vast web of theorems and conjectures connecting objects in number theory, representation theory, analysis and geometry. Central to the Langlands program are automorphic forms and their associated L-functions. These objects arise from analysis and representation theory. Despite slow and steady progress, many fundamental questions about the relationship between automorphic forms and number theory/arithmetic geometry remain unanswered. The goal of this project is gain new insights into some of these questions by making a deep investigation of three key topics: a) Nearly holomorphic Siegel modular forms, b) Deligne's conjecture on algebraicity of critical L-values, c) Ratios of Petersson norms for functorially related automorphic forms.Nearly holomorphic Siegel modular forms were first introduced by Shimura and have been indispensable for studying special values of automorphic L-functions. However, despite their ubiquity, their arithmetic properties and place in the Langlands framework have not yet been fully understood. This project will study their representation-theoretic and arithmetic properties and prove a close link between them and "vector valued Siegel modular forms".The insights gained from the study of nearly holomorphic modular forms will be used to tackle some special cases of a famous conjecture made by the Fields medal winning mathematician Pierre Deligne. The simplest example of Deligne's conjecture is the classical fact that the value of the Riemann zeta function at all positive even integers is a power of pi times a rational number. This research will prove the expected rationality results for much more complicated L-functions that are of great importance in several fields of mathematics. This research will also aim to prove a well-known conjecture made by Ibukiyama and Katsurada concerning ratios of Petersson norms for certain Siegel modular forms. This problem has deep significance as it concerns the behavior of arithmetic properties of automorphic forms under Langlands lifting. Solving this will require innovative adaptation to existing methods.The methods used in this project will be a powerful mix of classical techniques, modern representation theoretic methods, and group cohomology. This research will open several new avenues for further exploration.