Chomology of Arithmetic Groups and Galois Representations

算术群和伽罗瓦表示的系统学

• 批准号: 0139287
• 负责人: Avner Ash
• 金额: $ 9.1万
• 依托单位: Boston College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-09-01 至 2005-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0139287&HistoricalAwards=false
• 关键词: Chomology; Arithmetic; Groups; Galois; Representations

The principal investigator, together with various collaborators, studiesthe cohomology of arithmetic groups as modules over an algebra of Heckeoperators. He considers conjectures connecting (1) mod-p cohomologyclasses of congruence groups which are eigenclasses for all the Heckeoperators with (2) mod-p representations of the Galois (symmetry) groupof the algebraic numbers. These conjectures amount to saying there shouldexist reciprocity laws (usually non-abelian) connecting the two types ofobjects. The conjectures are tested computationally, which leads tofurther refinements of them. The principal investigator outlines anapproach for proving them in some special cases. Another set of computercalculations probe for the existence of automorphic representations ofcohomological type which are not lifted from smaller rank groups. Theseare conjecturally connected with reciprocity laws of Langlands type.Finally, the investigator and a colleague develop a theory of p-adicfamilies of not necessarily p-ordinary cohomology classes for congruencegroups. They use this to study the question whether non-liftedautomorphic representations are "p-adically rigid". Algebraic number theory studies the properties of polynomialfunctions and equations whose coefficients are whole numbers. Ever sinceDescartes invented analytic geometry in the early 1600's, the ability tosolve algebraic equations has been central to mathematics, science andengineering. In the last 50 years a new set of applications of numbertheory has opened up in the field of cryptography. The principalinvestigator studies delicate questions involving the fine structure andproperties of solutions to certain systems of algebraic equations.Although one can often obtain approximate solutions using computers, thereare many very subtle and difficult questions concerning the existence andform of the exact solutions. There are a number of conjectures on thetable to explain what is going on, and the principal investigator hascontributed some of them. He and his colleagues study these conjecturesby verifying them in some cases by extensive computer calculations, and byproving them in special cases. At the core of this research is thesurprising connection with geometric objects ("cohomology classes") thatcan be constructed in spaces of n-dimensional crystal lattices.

主要研究者与各个合作者一起研究算术群的上同调作为 Hecke 算子代数上的模。 他考虑将 (1) 同余群的 mod-p 上同调类与 (2) 代数数的伽罗瓦（对称）群的 mod-p 表示连接起来的猜想，这些同余群是所有 Hecke 算子的本征类。 这些猜想等于说应该存在连接两类对象的互易律（通常是非交换律）。 这些猜想经过计算测试，从而进一步完善它们。 主要研究者概述了在某些特殊情况下证明它们的方法。 另一组计算机计算探测上同调类型的自同构表示的存在性，这些表示不是从较小的秩组中提取出来的。 据推测，这些与 Langlands 类型的互反律有关。最后，研究者和一位同事发展了同余群不一定是 p-普通上同调类的 p-adic 族理论。 他们用它来研究非提升自同构表示是否是“p 绝对刚性”的问题。代数数论研究系数为整数的多项函数和方程的性质。 自从笛卡尔在 1600 年代初期发明解析几何以来，求解代数方程的能力一直是数学、科学和工程的核心。 在过去的 50 年里，数论在密码学领域开辟了一系列新的应用。 首席研究员研究涉及某些代数方程组的精细结构和解的性质的微妙问题。虽然人们通常可以使用计算机获得近似解，但关于精确解的存在和形式存在许多非常微妙和困难的问题。有许多猜想可以解释正在发生的事情，首席研究员也贡献了其中的一些猜想。 他和他的同事们研究这些猜想，在某些情况下通过大量的计算机计算来验证它们，并在特殊情况下证明它们。 这项研究的核心是与可以在 n 维晶格空间中构造的几何对象（“上同调类”）的惊人联系。