Lattice map for Anderson t-motives: First approach

There exists a lattice map from the set of pure uniformizable Anderson t-motives to the set of lattices. It is not known what is the image and the fibers of this map. We prove a local result that sheds the first light to this problem and suggests that maybe this map is close to 1-1. Namely, let M(0) be a t-motive of dimension n and rank r=2n - the n-th power of the Carlitz module of rank 2, and let M be a t-motive which is in some sense “close” to M(0). We consider the lattice map M↦L(M), where L(M) is a lattice in C∞n. We show that the lattice map is an isomorphism in a “neighborhood” of M(0). Namely, we compare the action of monodromy groups: (a) from the set of equations defining t-motives to the set of t-motives themselves, and (b) from the set of Siegel matrices to the set of lattices. The result of the present paper gives that the size of a neighborhood, where we have an isomorphism, depends on an element of the monodromy group. We do not know whether there exists a universal neighborhood. Method of the proof: explicit solution of an equation describing an isomorphism between two t-motives by a method of successive approximations using a version of the Hensel lemma.