I’ve uploaded my first paper to the arxiv: http://arxiv.org/pdf/1605.02717v1.pdf

I construct a bunch of integral models for well known objects, mainly building on Kisin’s first integral models paper. Firstly I extend his models for Shimura varieties to models defined over rings like $\mathbb{Z}[1/N]$ (his are over rings more like $\mathbb{Z}_{(p)}$)

Secondly, and much more substantially, I construct integral models for what Milne calls the “Standard Principal Bundle”, which is a gadget that in particular allows one to construct automorphic vector bundles, but also to have a notion of “de Rham sheaves” and (with a little more work: another paper is on the way) (iso)crystals.

A useful vague way to think about such objects is the following. A Shimura variety is a moduli space of Tannakian subcategories of a certain shape (roughly that they look like Rep G) of a category slightly larger than the category of motives, together with a trivialization of the etale cohomology fibre functors (the “level structure”). The level structure is needed because otherwise one gets some kind of Artin stack it’s difficult to work with. The Standard Principal Bundle parameterizes the same thing but also throws in a trivialisation of the de Rham cohomology fibre functor, making it a G-torsor over the Shimura variety.

Of course, there are technical issues with this description, but it gives you a good idea of how the construction should go: essentially I handle the Hodge type case by systematically exploiting the de Rham cohomology of the universal abelian variety. I handle the case where G is a torus by using the theory of Kisin modules and their relation to lattices in de Rham cohomology (to get a uniqueness statement), and the theory of CM motives (to get existence). I then do some group theoretic gymnastics to pass from the Hodge and torus type cases to the general abelian type case.

Please let me know if you find any mistakes, and I hope you enjoy the paper. 🙂