Abstract: | In this thesis I study various incarnations of rational homotopy theory in the world of arithmetic geometry. In particular, I study unipotent crystalline fundamental groups in the relative setting, proving that for a smooth and proper family of geometrically connected varieties f:X->S in positive characteristic, the rigid fundamental groups of the fibres X_s glue together to give an affine group scheme in the category of overconvergent F-isocrystals on S. I then use this to define a global period map similar to the one used by Minhyong Kim to study rational points on curves over number fields.
I also study rigid rational homotopy types, and show how to construct these for arbitrary varieties over a perfect field of positive characteristic. I prove that these agree with previous constructions in the (log-)smooth and proper case, and show that one can recover the usual rigid fundamental groups from these rational homotopy types. When the base field is finite, I show that the natural Frobenius structure on the rigid rational homotopy type is mixed, building on previous results in the log-smooth and proper case using a descent argument.
Finally I turn to l-adic étale rational homotopy types, and show how to lift the Galois action on the geometric l-adic rational homotopy type from the homotopy category Ho(Q_l-dga) to get a Galois action on the dga representing the rational homotopy type. Together with a suitable lifted p-adic Hodge theory comparison theorem, this allows me to define a crystalline obstruction for the existence of integral points. I also study the continuity of the Galois action via a suitably constructed category of cosimplicial Q_l-algebras on a scheme. |