Overconvergent cohomology of higher rank groups

高阶群的过收敛上同调

• 批准号: 0701153
• 负责人: Robert Pollack
• 金额: $ 15.9万
• 依托单位: Trustees of Boston University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0701153&HistoricalAwards=false
• 关键词: Overconvergent; cohomology; higher; rank; groups

In the last five years, a tremendous amount of progress has been made in the p-adic theory of modular forms. Skinner and Urban's proof of the main conjecture, Khare and Winterberger's proof of Serre's conjecture, Emerton's and Kisin's independent work on the Fontaine-Mazur conjecture represent only a partial list of the incredible recent progress that has occurred. Moreover, there is a rich emerging p-adic theory of automorphic forms generalizing the corresponding p-adic theory of modular forms. However, the vast computational methods available in the study of classical modular forms, do not yet exist in the higher rank case. Currently, there is a gap in the understanding of the shape and structures of many of the fundamental objects that appear in this more general setting. Pollack proposes to make a concrete and explicit study of overconvergent cohomology of higher rank groups along the lines that he and Glenn Stevens did in the case of classical modular forms. This previous study had both theoretical and computational aspects that, in particular, led to a deeper understanding of two variable p-adic L-functions and to a p-adic method of computing (conjecturally) global points on elliptic curves. He hopes that a similar study in the higher rank case will lead to a better understanding of analogous phenomena in this more general setting. For over a hundred years, mathematicians have studied modular forms which are certain complex analytic functions possessing many symmetries. The definition of these objects is through analysis (the rigorous study of calculus); nonetheless, these objects have made their way into the center of number theory as they possess a wealth of arithmetic information of the type that would have interested Euclid, Gauss, Fermat, etc. A deep understanding of these arithmetic properties of modular forms was one of the key steps in Andrew Wiles' proof of Fermat's Last Theorem. Moreover, the theory of automorphic forms -- a far-reaching generalization of modular forms -- has proven to also be extremely rich arithmetically. However, unlike classical modular forms where one can compute with relative ease, general automorphic forms can be extremely difficult to compute and experiment with numerically. Pollack proposes to make an in depth study of the computation of certain kinds of automorphic forms with the hope of gaining a better arithmetic understanding of them through conjecture and numerical exploration.

在过去的五年里，模形式的 p-adic 理论取得了巨大的进步。 斯金纳和厄本对主要猜想的证明、卡雷和温特伯格对塞尔猜想的证明、埃默顿和基辛对方丹-马祖尔猜想的独立工作仅代表了最近所取得的令人难以置信的进展的一部分。 此外，还有丰富的新兴自同构形式 p-adic 理论，概括了相应的模形式 p-adic 理论。 然而，在经典模形式研究中可用的大量计算方法在更高阶的情况下尚不存在。 目前，对于出现在这种更普遍的环境中的许多基本物体的形状和结构的理解存在差距。 波拉克提议按照他和格伦·史蒂文斯在经典模形式的情况下所做的那样，对高阶群的过收敛上同调进行具体而明确的研究。 先前的研究具有理论和计算方面的优势，特别是，加深了对两个变量 p 进 L 函数的理解，并得出了计算（推测）椭圆曲线上全局点的 p 进方法。 他希望在更高级别的案例中进行类似的研究将有助于更好地理解这种更普遍的情况下的类似现象。一百多年来，数学家们一直在研究模形式，模形式是某些具有许多对称性的复杂解析函数。这些对象的定义是通过分析（微积分的严格研究）；尽管如此，这些对象已经进入了数论的中心，因为它们拥有丰富的算术信息，这些信息会引起欧几里得、高斯、费马等人的兴趣。深入理解模形式的这些算术性质是其中之一安德鲁·威尔斯证明费马大定理的关键步骤。 此外，自同构理论——模形式的一种影响深远的概括——已被证明在算术上也极其丰富。 然而，与可以相对容易地计算的经典模形式不同，一般自同构形式可能极其难以进行数值计算和实验。 波拉克建议对某些自守形式的计算进行深入研究，希望通过猜想和数值探索获得对它们更好的算术理解。