Models over Z for locally quasi-split algebraic groups.

摘要

Given an algebraic group G one can define a model over Z to be a smooth group scheme, G&barbelow;, whose general fiber, G⊗Q , is equal to G, and whose special fibers G⊗Z/pZ are reductive for all primes p∈Z . Only certain algebraic groups admit models over Z . In this thesis we introduce a modification to the definition of a model over Z allowing the consideration of models over Z for several new types of algebraic groups.;The thesis is divided into three parts. In part one we discuss the modified model definition and examine the types of algebraic groups which admit models under the new definition. In part two we adopt a counting argument using mass formulas related to certain double coset spaces. This allows us to compute the number of independent models which must exist for particular types of algebraic groups. At the end of part two we list results giving the number of models for groups of type A2n2,D1n +12 and E62 . In the last part of the thesis, we exhibit models for certain special unitary groups of rank less than or equal to 5.