Frobenius actions on the de Rham cohomology of Drinfeld modules

We study the action of endomorphisms of a Drinfeld A-module φ on its de Rham cohomology HDR(φ,L) and related modules, in the case where φ is defined over a field L of finite Acharacteristic p. Among others, we find that the nilspace H0 of the total Frobenius FrDR on HDR(φ,L) has dimension h = height of φ. We define and study a pairing between the p-torsion pφ of φ and HDR(φ,L), which becomes perfect after dividing out H0. Introduction. The theory of Drinfeld modules (introduced in 1974 by V. G. Drinfeld as “elliptic modules” [4]) forms the core of the modern arithmetic of function fields. Work of many researchers (see the bibliography in [11] for an early account) contributes to establish deep results about Drinfeld modules and their moduli theory and connections with e.g. automorphic forms, Galois representations, the Langlands program, arithmetic groups, abelian varieties, transcendence theory, as well as to various applications. One feature is the existence of “cohomology theories”, which associate to each Drinfeld A-module φ over a suitable A-field L (see sect. 1 for definitions and requirements) vector spaces analogous with the Betti, the l-adic, and the de Rham (co-)homology of an abelian variety. Rightly speaking, these are only first (co-)homology modules in the Drinfeld module framework (so we don’t dispose of “true” cohomology theories), but these are provided with all the structure (functoriality, comparison isomorphisms, GAGA-type theorems, formalism of vanishing cycles) expected from the analogy with the first cohomologies of abelian varieties [4, 3, 7, 8]. In the present paper, we continue the study of the de Rham module HDR(φ, L) in the case where the field L of definition of φ has finite A-characteristic. Here we dispose of different Frobenius actions (geometric, arithmetic, total Frobenius) on HDR(φ, L) and related modules. We prove two basic results: