Arithmetic Algebraic Geometry

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Abstract for Award DMS-0400482 "Arithmetic algebraic Geometry"by Ching-Li ChaiThis project is in the field of arithmetic algebraic geometry and contains two parts: geometry of Shimura varieties and Neron models of semiabelian varieties. The first part of this project is centered around the Hecke orbit conjecture for good reductions of Shimura variety, formulated by Oort, which states that the Zariski closure of a prime-to-p Hecke orbit is equal to the Zariski closure of a leaf. Partly in collaboration with his collaborators, F. Oort and C.-F. Yu, Chai has developed several techniques toward the Hecke orbit conjecture, and also formulated a plan to prove the Hecke orbit conjecture for the moduli space of abelian varieties. The last step of the plan was finished by C.-F. Yu, and an outline of a proof of the Hecke orbit conjecture for the moduli space of abelian varieties is available. A main objective of this proposal is a detailed exposition of that proof, as well as further development of the method for future applications. Also will be explored is the Hecke orbit problem for Shimura varieties attached to unitary groups. The focus of the second part is a numerical invariant of semiabelian varieties over local fields, called the base change conductor, defined using the Neron models. The goal here is to understand the behavior of the base change conductor when the residue field is not perfect, and to explore the foundational properties of formal Neron models of rigid analytic spaces.The first part of this project studies the of symmetries on a very special class of polynomial equations. This class of polynomial equations, called Shimura varieties, are of central importance in Number Theory. Conjecturally, these symmetries characterize a structure, called "foliation", on Shimura varieties. The proposal is to develop and document a recently conceived proof of this conjecture. The second part of this project deals with another aspect of system of polynomial equations, on the drop of the level of complexity of a system of equation when more general numbers are allowed to be used for solutions. This project is expected to enhance our knowledge in number theory, a subject which, though considered platonic and pure in the past, has found a plethora of applications in the digital age.