Abstract: We give a new proof of the Adams-Riemann-Roch theorem for a smooth projective morphism in the situation where the base scheme is of characteristic p > 0 and is regular and quasi-pro jective over a noetherian affine scheme. We also answer a question of B. Köck.

Abstract: We prove the isogeny conjecture for A-motives over finitely generated fields K of transcendence degree \le 1. This conjecture says that for any semisimple A-motive M over K, there exist only finitely many isomorphism classes of A-motives M' over K for which there exists a separable isogeny M' \to M. The result is in precise analogy to known results for abelian varieties and for Drinfeld modules and will have strong consequences for the ℘-adic and adelic Galois representations associated to M. The method makes essential use of the Harder-Narasimhan filtration for locally free coherent sheaves on an algebraic curve.

Abstract: Let φ be a Drinfeld A-module of arbitrary rank and generic characteristic over a finitely generated field K. If the endomorphism ring of φ over an algebraic closure of K is equal to A, we prove that the image of the adelic Galois representation associated to φ is open.

The Isogeny Conjecture for t-Motives Associated to Direct Sums of Drinfeld Modules (with Matthias Traulsen)
J. Number Theory 117 (2006), no. 2, 355-375.

Abstract: Let K be a finitely generated field of transcendence degree 1 over a finite field. Let M be a t-motive over K of characteristic ℘₀, which ist semisimple up to isogeny. The isogeny conjecture for M says that there are only finitely many isomorphism classes of t-motives M' over K, for which there exists a separable isogeny M' → M of degree not divisible by ℘₀. For the t-motive associated to a Drinfeld module this was proved by Taguchi. In this article we prove it for the t-motive associated to any direct sum of Drinfeld modules of characteristic ℘₀ different from 0.

On Weil restriction of reductive groups and a theorem of Prasad
Math. Z. 248 (2004), no. 3, 449-457.

Abstract: Let G be a connected simple semisimple algebraic group over a local field F of arbitrary characteristic. In a previous article by the author the Zariski dense compact subgroups of G(F) were classified. In the present paper this information is used to give another proof of a theorem of Prasad (also proved by Margulis) which asserts that, if G is isotropic, every non-discrete closed subgroup of finite covolume contains the image of G^~(F) where G^~ denotes the universal covering of G. This result played a central role in Prasad's proof of strong approximation. The present proof relies on some basic properties of Weil restrictions over possibly inseparable field extensions, which are also proved here.

Abstract: A theorem of Green, Lazarsfeld and Simpson (formerly a conjecture of Beauville and Catanese) states that certain naturally defined subvarieties of the Picard variety of a smooth projective complex variety are unions of translates of abelian subvarieties by torsion points. Their proof uses analytic methods. We refine and give a completely new proof of their result. Our proof combines galois-theoretic methods and algebraic geometry in positive characteristic. When the variety has a model over a function field and its Picard variety has no isotrivial factors, we show how to replace the galois-theoretic results we need by results from model theory (mathematical logic). Furthermore, we prove partial analogs of the conjecture of Beauville and Catanese in positive characteristic.

Abstract: Let C be a complete non-archimedean valued algebraically closed field of characteristic p>0 and consider the punctured unit disc D in C. Let q be a power of p and consider the arithmetic Frobenius automorphism σ: D → D, x |→ x^{q -1} . A sigma-bundle is a vector bundle F on D together with an isomorphism σ^*F → F. The aim of this article is to develop the basic theory of these objects and to classify them. It is shown that every σ-bundle is isomorphic to a direct sum of certain basic σ-bundles F_d,r which depend only on rational numbers d/r. This result has close analogies with the classification of rational Dieudonne modules and of vector bundles on the projective line or on an elliptic curve. It has interesting consequences concerning the uniformizability of Anderson's t-motives that will be treated in a future paper.

Abstract: Let A be a semiabelian variety over an algebraically closed field of arbitrary characteristic, endowed with a finite morphism ψ: A → A. In this paper we give an essentially complete classification of all ψ-invariant subvarieties of A. For example, under some mild assumptions on (A,ψ) we prove that every ψ-invariant subvariety is a finite union of translates of semiabelian subvarieties. This result is then used to prove the Manin-Mumford conjecture in arbitrary characteristic and in full generality. Previously, it had been known only for the group of torsion points of order prime to the characteristic of K. The proofs involve only algebraic geometry, though scheme theory and some arithmetic arguments cannot be avoided.

Abstract: The Manin-Mumford conjecture in characteristic zero was first proved by Raynaud. Later, Hrushovski gave a different proof using model theory. His main result from model theory, when applied to abelian varieties, can be rephrased in terms of algebraic geometry. In this paper we prove that intervening result using classical algebraic geometry alone. Altogether, this yields a new proof of the Manin-Mumford conjecture using only classical algebraic geometry.

Abstract: Let X be an irreducible smooth projective curve over an algebraically closed field of characteristic p>0. Let k be either a finite field of characteristic p or a local field of residue characteristic p. Let F be a constructible etale sheaf of k-vector spaces on X. Suppose that there exists a finite Galois covering π: Y → X such that the generic monodromy of π*F is pro-p and Y is ordinary. Under these assumptions we derive an explicit formula for the Euler-Poincaré characteristic c(X,F) in terms of easy local and global numerical invariants, much like the formula of Grothendieck-Ogg-Shafarevich in the case of different characteristic. Although the ordinariness assumption imposes severe restrictions on the local ramification of the covering π, it is satisfied in interesting cases such as Drinfeld modular curves.

Abstract: Consider a finitely generated Zariski dense subgroup Γ of a connected simple algebraic group G over a global field F. An important aspect of strong approximation is the question of whether the closure of Γ in the group of points of G with coefficients in a ring of partial adeles is open. We prove an essentially optimal result in this direction, based on the condition that Γ is not discrete in that ambient group. There are no restrictions on the characteristic of F or the type of G, and simultaneous approximation in finitely many algebraic groups is also studied. Classification of finite simple groups is not used.

Abstract: Generalizing a classical theorem of Jordan to arbitrary characteristic, we prove that every finite subgroup of GL_n over a field of any characteristic p possesses a subgroup of bounded index which is composed of finite simple groups of Lie type in characteristic p, a commutative group of order prime to p, and a p-group. While this statement can be deduced from the classification of finite simple groups, our proof is self-contained and uses methods only from algebraic geometry and the theory of linear algebraic groups. We believe that our results can serve as a viable substitute for classification in a range of applications in various areas of mathematics.

Abstract: 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 t-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.

Motives and Hodge Structures over Function Fields
Talk at the Arbeitstagung 1997 in Bonn, 6 p.