On the conjecture of Langlands and Rapoport.

The conjecture of Langlands and Rapoport describes, under certain hypotheses, the set of points on the reduction of a Shimura variety at a prime of the reflex field. The description is sufficiently group theoretic in nature so as to allow comparison between automorphic L-functions and the zeta function of the Shimura variety. Langlands and Rapoport assume the group used to define the Shimura variety splits over an unramified extension at the prime in question, and that the derived group is simply connected. Their conjecture is stated in the cases where the variety has good reduction (i.e., where the variety is defined using a hyperspecial subgroup), or when it has 'mild' bad reduction (i.e., where the adjoint group is anisotropic at the prime in question, and one uses the unique special group to define the Shimura variety). This thesis generalises the conjecture: arbitrary parahoric groups are allowed. The main application of the results in the thesis is the following. Suppose the centre of the reductive group used in the definition of the Shimura variety has trivial cohomology. Then, to prove the original conjecture for Shimura varieties with mild bad reduction, it suffices to prove a sufficiently functorial version of the generalised conjecture for Shimura varieties defined using a parahoric subgroup that is the intersection of hyperspecial subgroups. The method is to embed the variety with mild bad reduction into a second Shimura variety, and then to use a descent argument.PhDMathematicsPure SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/131428/2/9909892.pd