Analogy between Anderson T-motives and abelian varieties with multiplication by imaginary quadratic fields, and related problems

Anderson T-motives are the functional field analogs of
abelian varieties, but this analogy is rather weak. For example, the
analogs of dimension $g$ of an abelian variety are 2 numbers:
dimension $n$ and rank $r$. There exists a more strong analogy between
Anderson T-motives of dimension $n$ and rank $r$ and abelian varieties
of dimension $r$ with multiplication by an imaginary quadratic field
$K$ (MIQF), of signature $(n, r-n)$, because both objects have the
same Deligne group $GU(n, r-n)$.

This analogy permits us to get 2 results in the theory of abelian
varieties with MIQF. Firstly, we associate it a "$K$-lattice" of rank
$r$ in $C^n$ (recall that $n < r$ ). Secondly, for $n=1$ we can define
exterior powers of these varieties.

Finally, we consider some problems related to the theory of duality
for Anderson T-motives, and a problem whether we have a local 1 -- 1
correspondence between pure uniformizable Anderson T-motives, and
lattices in $C_\infty^n$ -- all in a neighborhood of a distinguished
point (subject in development).