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.

主要研究者（PI），与各种合作者，研究一个领域的数论有关的表示伽罗瓦集团overthe有理数和上同调的有限指数子群GL（n，Z）。 Thelink之间的这些是由行动的Hecke运营商上同调和图像的Frobenius元素在伽罗瓦表示。 PI探索了“ADPS”猜想，其中上同调和伽罗瓦表示都是mod-p值的。 他outlinesan方法证明这一猜想在某些3维的情况下，测试它在某些4维的情况下，并独立verifyingsome其后果丢番图问题。 PI提出了一个新的猜想，把mod-p对象与自守表示联系起来，并给出了在三维情况下证明这个猜想的方法。 最后，PI和一个同事继续研究自守上同调的p-adicfamilies。 一方面，对于某些类，例如GL（3）上的类，而不是从GL（2）上提升的类，他们考虑p-adic刚性的问题。 另一方面，对于可变形类，他们研究了2-变元p-adic L-函数。 它也一直是数学中的主要主题之一，既受到这些应用的驱动，也受到其内在美学吸引力的驱动。 当方程具有整数系数时，对它们的研究就成为数论的一部分。 素数理论（除了1之外没有更小因子的数，如2、3、5和7）是这里的核心。 在过去的50年里，这些理论的应用已经成为密码学和通信理论的关键。首席研究员研究了关于多项式方程组的解集的精细结构，它们表现出的对称性以及它们与各种有趣的几何和拓扑对象的关系（通常令人惊讶）的微妙问题。 这些关系是由方程对不同素数的反应来调节的。 对于这些关系有许多理论上的解释，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。 他还调查的现象，比较整个家庭的thesestructures相对于一个固定的素数。