Uniformizers for Elliptic Sheaves

In this paper we define k-elliptic sheaves, A-motives and t-modules over A, which are obvious generalizations of elliptic sheaves, t-motives and t-modules. Following results of [An1], [D], [LRSt], [Mu], [St],... we shall obtain the equivalence of this objects. Bearing in mind [Al] we also describe a correspondence between k-elliptic sheaves with formal level structures (2.9) and discrete subspaces. The moduli scheme for these objects shall be a subscheme of Sato’s infinite Grassmannian, in this setting the moduli for classical t-motives will be a closed subscheme of this last subscheme. In the section 6 for these discrete subspaces, we show a result analogous to [An1] for the behaviour of the determinant. In the same way A-motives with formal level structures have associated locally dense subspaces (3.1) these subspaces are determined by a subspace of formal series of dimension k, called subspace of uniformizers (3.5). With this we see that a kelliptic sheaf with formal level structure could be seen as a orbit of the action of Gln(Fq[[tx]]) over the set of these subspaces of uniformizers. These subspaces allow us to get an injective morphism between the moduli scheme of k-elliptic sheaves with formal level structures and a Grassmannian of finite k-dimensional subspaces of formal series, Definition 3.3. An explicit computation can be made for these subspaces of uniformizers for k = 1 and one can check certain analogy between these subspaces of uniformizers and Dirichlet series for Drinfeld modules [G]. These uniformizers allow us to obtain the arithmetic counterpart of the Baker function, [SW], defined in the setting of the theory of soliton equations (see [BlS] for elliptic sheaves together an introduction of soliton theory, an intensive study for 1-elliptic modules is done in [An2]). We can also make explicit the action of the classical