Hodge Structures over Function Fields

We develop a general theory of mixed Hodge structures over local or global function fields which in many ways resembles the formalism of classical Hodge structures. Our objects consist of a finite dimensional vector space together with a weight filtration, but instead of a Hodge filtration we require finer information. In order to obtain a reasonable category we impose a semistability condition in the spirit of invariant theory and prove that the resulting category is tannakian. This allows us to define and analyze Hodge groups and determine them in some cases. The analogies with classical mixed Hodge structures range from the role of semistability to the fact that both objects arise from the analytic behavior of motives. The precise relation of our objects with the analytic uniformization of Anderson's -motives will be the subject of a separate paper. For Hodge structures arising from Drinfeld modules we can combine the present results with a previous one on Galois representations, obtaining a precise analogue of the Mumford-Tate conjecture