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

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"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.