Let $M$ be an Anderson T-motive (the simplest case, i.e. abelian in the meaning of [G] Goss, Basic structures of function field arithmetic, Def. 5.4.12, over $A^1$, having $N=0$), and let $H_1(E)$ ([G], 5.9.11.3) be its lattice - more exactly, the lattice of the associated T-module $E$, and $H^1(M):=M\{T\}^\tau$ from [G], 5.9.11.2.
If $M$ is uniformizable then there exists a perfect pairing between $H_1(E)$ and $H^1(M)$. If $M$ is not uniformizable then there exists a map $H_1(E) \to Hom \ ( H^1(M), \Bbb F_q[T] )$. What is known on it: is it an iso-, epi-, monomorphism, or none of these properties?
