Points of Integral Models of Shimura Varieties of Hodege Type and the Tate and Langlands--Rapoport Conjectures

“这项奖励是根据2009年美国复苏和再投资法案（公法111-5）资助的。”本课题研究了具有良好积分模型的志村型阿贝尔型特种纤维。特殊纤维是光滑的，有限域上的准投影品种，由于它们的预期（推测）模解释而具有丰富的结构。许多悬而未决的问题和猜想都与它们有关。该项目将调查与特殊纤维有关的三个主要领域。首先，它将研究具有晶体性质的特殊纤维的不同分层（如合理分层、m级分层和特拉弗索分层）。其次，它将研究与有限域上有值的点相关的动机，以便在某些情况下证明关于有限域上阿贝尔变上的代数循环的经典Tate猜想的一个椭圆版本。第三，它将计算特殊纤维有限域中有值的点，以便在许多情况下证明朗兰兹—拉波波特的一个具有组合性质的猜想。求解有限域上的多项式方程组是数论中的一个中心问题。当解具有动机（模）解释时，人们可以将许多解析和组合性质的新对象与方程组联系起来，这些对象的性质与方程组的几何和算术性质密切相关。本课题旨在求解和研究具有额外结构的参数化阿贝尔变量的有限域方程组。这项研究导致了几何学、组合学和分析学之间的基本相互作用。例如，解可以自然地组合在一起，以定义可用于描述（计数）解和研究与解自然相关的阿贝尔变量（动机）的方程系统的不同层次。

"This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5)."This project investigates special fibres of good integral models of Shimura varieties of abelian type.The special fibres are smooth, quasi-projective varieties over finite fields which have a rich structure due to their expected (conjectured) moduli interpretation.Many open problems and conjectures pertain to them.The project will investigate three main areas pertaining to special fibres.First it will study different stratifications of special fibres that are of crystalline nature (like the rational stratifications, the level m stratifications, and the Traverso stratifications).Second it will study the motives associates to points with values in finite field in order to prove in some cases an ad\'elic version of a classical conjecture of Tate pertaining to algebraic cycles on abelian varieties over finite fields.Third it will count the points with values in finite fields of special fibres in order to prove in many cases a conjecture of Langlands--Rapoport which is of combinatorial nature.Solving polynomial systems of equations over finite fields is a central problem in number theory.When the solutions have a motivic (moduli) interpretation, one can associate to the system of equations many new objects of analytic and combinatorial nature whose properties are very much interrelated to the geometric and arithmetic properties of the system of equations.The project aims at solving and studying those systems of equations over finite fields which parametrize abelian varieties endowed with extra structure. The study leads to a fundamental interplay between geometry, combinatorics, and analysis.For instance, the solutions can be naturally grouped together to define different stratifications of the systems of equations that can be used to describe (count) the solutions and to study the abelian varieties (motives) associated naturally to the solutions.