Non-injectivity of the lattice map for non-mixed Anderson t-motives, and a result towards its surjectivity

Let [Formula: see text] be an uniformizable Anderson t-motive and [Formula: see text] its lattice. First, we prove by an explicit construction that for the non-mixed [Formula: see text], the lattice map [Formula: see text] is not injective. Second, we show that [Formula: see text] lattices [Formula: see text] such that [Formula: see text] for pure [Formula: see text], but [Formula: see text] a non-pure [Formula: see text] such that [Formula: see text]. This is a result towards surjectivity of the lattice map. The t-motives used in the proofs are non-pure t-motives of dimension 2, rank 3. Finally, we start calculations in order to answer a question whether all non-pure t-motives of dimension 2, rank 3 are uniformizable, or not.