FRG: L-functions and Modular Forms

FRG：L 函数和模块化形式

• 批准号: 0757627
• 负责人: William Stein
• 金额: $ 120.53万
• 依托单位: American Institute of Mathematics
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2012-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0757627&HistoricalAwards=false
• 关键词: FRG; functions; Modular; Forms

The PI and his research team are proposing a major new project to develop theory and organize methods for understanding and computing with L-functions and modular forms. Broadly speaking, they plan to chart the landscape of L-functions and modular forms in a systematic and concrete fashion. They will study these functions, develop algorithms for their computation, and test fundamental conjectures, including: the Generalized Riemann Hypothesis, the Birch and Swinnerton-Dyer conjecture, the Bloch-Kato conjecture, the correlation conjectures of Montgomery and of Rudnick-Sarnak, the density conjectures of Katz-Sarnak, automorphy of the Hasse-Weil zeta functions, and the Selberg eigenvalue conjecture. They plan to carry out a systematic study, theoretically, algorithmically, and experimentally of degree 1, 2, 3, 4 L-functions and their associated modular forms, including classical modular forms, Maass forms for GL(2), GL(3), GL(4), Siegel modular forms, and Hilbert modular forms. They will also investigate symmetric square and cube L-functions, Rankin-Selberg convolution L-functions, the Hasse-Weil L-functions of algebraic varieties, Artin L-functions associated to 3- and 4-dimensional Galois representations, and, less systematically, look at a few high degree L-functions associated to higher symmetric powers and higher dimensional Galois representations.L-functions and modular forms underlie much of twentieth century number theory and are connected to the practical applications of number theory in cryptography. Virtually all branches of number theory have been touched by L-functions and modular forms. Besides containing deep information concerning the distribution of prime numbers and the structure of elliptic curves, they feature prominently in Andrew Wiles' solution of the famous 350-year-old Fermat's Last Theorem, and in the twentieth century classification of congruent numbers, a problem first posed by Arab mathematicians one thousand years. In spite of their central importance, mathematicians have only scratched the surface of these crucial and powerful functions. The PI and his research team are undertaking a major new project to systematically tabulate and study these functions. Their work will fall into four categories: theoretical, algorithmic, experimental, and data gathering. The theoretical work will be stimulated by their goal of charting the world of L-functions and modular forms. Their experimental work will involve testing many key conjectures concerning these functions. The project will produce a large amount of training, with plans for three graduate student schools, an undergraduate research experience, and support for a score of postdocs and graduate students who will assist in research. It will result in the creation of a vast amount of data about a wide range of modular forms and L-functions, which will far surpass in range and depth anything computed before in this area. The data will be organized in a freely available online data archive, along with the actual programs that were used to generate these tables. By providing these tables and tools online, the researchers will guarantee that the usefulness of this project will extend far beyond the circle of researchers on this FRG. The archive will be a rich source of examples and tools for researchers working on L-functions and modular forms for years to come, and will allow for future updates and expansion.

PI和他的研究团队正在提出一个重大的新项目，以发展理论，并用L组织理解和计算的方法-函数和模块形式。概括地说，他们计划以系统和具体的方式绘制L的景观-功能和模块形式。他们将研究这些函数，开发其计算算法，并测试基本猜想，包括：广义Riemann假设，Birch和Swinnerton-Dyer猜想，Bloch-Kato猜想，Montgomery和Rudnick-Sarnak的相关猜想，Katz-Sarnak的密度猜想，Hasse-Weil Zeta函数的自同构，以及Selberg本征值猜想。他们计划从理论上、算法上和实验上对1、2、3、4次L函数及其相关的模形式进行系统的研究，包括经典模形式、GL(2)、GL(3)、GL(4)的Maass形式、Siegel模形式和Hilbert模形式。他们还将研究对称的正方和立方L函数，兰金-塞尔伯格卷积L-函数，代数族的哈斯-韦尔-L-函数，与三维和四维伽罗瓦表示有关的阿尔廷L-函数，以及较不系统地研究几个与更高对称幂和高维伽罗瓦表示有关的高次L-函数。L-函数和模形式是20世纪数论的基础，并与数论在密码学中的实际应用有关。L几乎触及了数论的所有分支--函数和模形式。除了包含有关素数分布和椭圆曲线结构的深入信息外，它们在Andrew Wiles的著名的350年前的费马最后定理的解决方案中占有突出地位，在20世纪的同余数分类中也是如此，这是一个由阿拉伯数学家在1000年前首次提出的问题。尽管它们至关重要，但数学家们只是触及了这些关键而强大的函数的皮毛。PI和他的研究团队正在进行一项新的重大项目，以系统地将这些功能制表和研究。他们的工作将分为四类：理论、算法、实验和数据收集。他们的目标是描绘L的世界--函数和模块形式，这将刺激理论工作。他们的实验工作将包括测试与这些函数有关的许多关键猜想。该项目将产生大量的培训，计划开设三所研究生院，一次本科生研究经历，并支持数十名博士后和研究生协助研究。它将导致创建关于广泛的模块形式和L函数的大量数据，其范围和深度将远远超过该领域以前计算的任何数据。这些数据将被组织在一个免费可用的在线数据档案中，以及用于生成这些表格的实际程序。通过在网上提供这些表格和工具，研究人员将保证这个项目的有用性将远远超出这个FRG的研究人员的圈子。在未来的几年里，该档案将为研究L函数和模块化形式的研究人员提供丰富的示例和工具来源，并将允许未来的更新和扩展。