FRG: L-functions and Modular Forms

PI和他的研究团队正在提出一个重大的新项目，以发展理论和组织方法来理解和计算l函数和模块形式。一般来说，他们计划以系统和具体的方式绘制l函数和模块形式的景观图。他们将研究这些函数，为它们的计算开发算法，并检验基本猜想，包括：广义黎曼猜想，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特征值猜想。他们计划从理论上、算法上和实验上对1、2、3、4阶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方形和立方体l函数，Rankin-Selberg卷积l函数，代数变体的Hasse-Weil l函数，与3维和4维伽罗瓦表示相关的Artin l函数，并且，不太系统地研究与更高对称幂和高维伽罗瓦表示相关的一些高阶l函数。l函数和模形式是20世纪数论的基础，并且与数论在密码学中的实际应用有关。实际上数论的所有分支都涉及到l函数和模形式。除了包含关于素数分布和椭圆曲线结构的深刻信息外，它们在安德鲁·怀尔斯（Andrew Wiles）对350年前著名的费马大定理（Fermat’s Last Theorem）的解答，以及在20世纪对同余数的分类（一千年前阿拉伯数学家首次提出的问题）中发挥了重要作用。尽管它们的核心重要性，数学家们只触及了这些关键和强大的函数的表面。PI和他的研究小组正在进行一个重大的新项目，系统地制表和研究这些功能。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函数和模形式的世界，这将刺激他们的理论工作。他们的实验工作将包括检验关于这些功能的许多关键猜想。该项目将产生大量的培训，计划建立三个研究生学院，本科生研究经验，并支持数十名博士后和研究生协助研究。它将导致创建大量关于各种模块形式和l函数的数据，其范围和深度将远远超过该领域以前计算过的任何数据。这些数据将与用于生成这些表的实际程序一起组织在一个免费的在线数据存档中。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。该档案将成为未来几年研究l函数和模块化形式的研究人员的示例和工具的丰富来源，并将允许未来的更新和扩展。

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.