Cohomology of Arithmetic Groups and Galois Representations

主要研究者（PI）与许多合作者，研究了数论的一个领域，涉及GL（n，Z）有限指数子群的有理数上伽罗瓦群的表示和上同调。它们之间的联系是由Hecke算子对上同调的作用和伽罗罗伊斯表示中Frobenius元素的象给出的。PI探索了“ADPS”猜想，其中的同调和伽罗瓦表示都是模p值。他概述了在某些三维情况下证明这个猜想的方法，在某些四维情况下测试它，并独立验证了它对丢番图问题的一些结果。PI提出了一个新的猜想，将模p对象与自同构表示联系起来，并提出了在三维情况下证明这个猜想的方法。最后，PI和他的同事继续他们对自同构上同的p-adicfamilies的研究。一方面，对于某些班级，e。对于GL(3)上的那些没有脱离GL(2)的人，他们考虑的是元刚性问题。另一方面，对于可变形类，他们研究了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。它也一直是数学的主要主题之一，由这些应用和其内在的美学吸引力驱动。当方程具有整数系数时，它们的研究就成为数论的一部分。质数理论（除1以外没有更小的因数的数，如2、3、5和7）是本书的核心。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。首席研究员研究有关多项式方程系统的解集的精细结构，它们所表现出的对称性以及它们与各种有趣的几何和拓扑对象（通常令人惊讶）的关系的微妙问题。这些关系是由方程相对于各种素数的表现所介导的。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，在更复杂的情况下用计算机验证它们。他还研究了将这些结构的整个族与一个固定素数进行比较的现象。

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.