Integral canonical models for G-bundles on Shimura varieties of abelian type

Abstract

This thesis builds on Kisin's theories of S-modules and integral models for Shimura varieties of abelian type to further our understanding of the arithmetic of Shimura varieties in several directions. First, we show that Shimura varieties ShK(G;X) of abelian type with level K hyperspecial at primes away from N and reflex field E admit canonical smooth integral models over OE[1/N]. Next, under the condition that Z(G) is split by a CM field, we prove that the standard principal G-bundles PK(G;X) also admit canonical integral models over OE[1/N], which are characterised uniquely using the theory of S-modules, and these in turn give rise to integral models for automorphic vector bundles. Finally, working over p-adic completions of Kisin's models, and dropping the restriction on Z(G) by working more purely with p-adic Hodge theory, we construct similarly canonical families of filtered F-crystals with G-structure. We then use these to deduce that the Galois representations arising in the p-adic cohomology of Shimura varieties with nontrivial coefficient sheaves are crystalline when the level is hyperspecial at p in the case of proper Shimura varieties of abelian type.