Cohomology of Arithmetic Groups and Galois Representations

算术群的上同调和伽罗瓦表示

• 批准号: 0455240
• 负责人: Avner Ash
• 金额: --
• 依托单位: Boston College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-09-01 至 2008-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0455240&HistoricalAwards=false
• 关键词: Cohomology; Arithmetic; Groups; Galois; Representations

The Principal Investigator (PI), with various collaborators, studies anarea of number theory concerned with representations of Galois groups overthe rationals and cohomology of finite-index subgroups of GL(n,Z). Thelink between these is given by the action of Hecke operators on thecohomology and the image of Frobenius elements in the Galoisrepresentation. The PI explores the "ADPS" conjecture, where thecohomology and Galois representation are both mod-p valued. He outlinesan approach for proving this conjecture in certain 3-dimensional cases,testing it in certain 4-dimensional cases, and independently verifyingsome of its consequences for Diophantine problems. The PI states a newconjecture that links the mod-p objects with automorphic representationsand suggests an approach for proving this conjecture in the 3-dimensionalcase. Finally, the PI and a colleague continue their study of p-adicfamilies of automorphic cohomology. On the one hand, for certain classes,e.g those on GL(3) not lifted from GL(2), they consider questions ofp-adic rigidity. On the other hand, for deformable classes, theyinvestigate 2-variable p-adic L-functions.The ability to solve systems of algebraic equations has been central tomodern science and engineering. It has also always been one of the maintopics in mathematics, driven both by these applications and by itsintrinsic aesthetic appeal. When the equations have whole-numbercoefficients, their study becomes part of number theory. The theory ofprime numbers (numbers without smaller divisors except 1, such as 2,3,5and 7) is central here. In the last 50 years, applications of thesetheories have become crucial to cryptography and communications theory.The Principal Investigator studies delicate questions concerning the finestructure of sets of solutions to systems of polynomial equations, thesymmetries they exhibit, and their (often surprising) relationship tovarious interesting geometric and topological objects. Theserelationships are mediated by how the equations behave with respect to thevarious prime numbers. There are a number of conjectural explanations ofthese relationships, and the PI studies them, proving them in certain"easy" cases and verifying them by computer in more complicated cases. Healso investigates phenomena which compare whole families of thesestructures with respect to a fixed prime number.

首席研究员 (PI) 与多位合作者一起研究数论的一个领域，该领域涉及 GL(n,Z) 有限指数子群的有理数和上同调上的伽罗瓦群的表示。 这些之间的联系是由赫克算子对上同调的作用和伽罗瓦表示中的弗罗贝尼乌斯元素的图像给出的。 PI 探索了“ADPS”猜想，其中上同调和伽罗瓦表示都是 mod-p 值。 他概述了在某些 3 维情况下证明该猜想、在某些 4 维情况下测试它并独立验证其对丢番图问题的一些后果的方法。 PI 陈述了一个新的猜想，它将 mod-p 对象与自同构表示联系起来，并提出了一种在 3 维情况下证明该猜想的方法。 最后，PI 和一位同事继续研究自同构的 p-adic 族。 一方面，对于某些类别，例如 GL(3) 上未从 GL(2) 提升的类别，他们考虑 p-adic 刚性问题。 另一方面，对于可变形类，他们研究 2 变量 p 进 L 函数。求解代数方程组的能力一直是现代科学和工程的核心。 由于这些应用及其内在的审美吸引力，它也一直是数学的主要主题之一。 当方程具有整数系数时，它们的研究就成为数论的一部分。 素数理论（除 1 之外没有更小的因数的数，例如 2、3、5 和 7）是这里的核心。 在过去的 50 年里，这些理论的应用对于密码学和通信理论已经变得至关重要。首席研究员研究了一些微妙的问题，涉及多项式方程组解集的精细结构、它们表现出的对称性，以及它们与各种有趣的（通常令人惊讶的）关系。几何和拓扑对象。 这些关系是由方程相对于各种素数的表现来调节的。 对这些关系有许多推测性的解释，PI 研究它们，在某些“简单”的情况下证明它们，并在更复杂的情况下通过计算机验证它们。 他还研究了将这些结构的整个家族与固定素数进行比较的现象。