$h^1\ne h_1$ for Anderson t-motives

Abstract

Let $M$ be an Anderson t-motive of dimension $n$ and rank $r$. Associated are two $\Bbb F_q[T]$-modules $H^1(M)$, $H_1(M)$ of dimensions $h^1(M)$, $h_1(M)\le r$ - analogs of $H^1(A,\Bbb Z)$, $H_1(A,\Bbb Z)$ for an abelian variety $A$. There is a theorem (Anderson): $h^1(M)=r \iff h_1(M)=r$; in this case $M$ is called uniformizable. It is natural to expect that always $h^1(M)=h_1(M)$. Nevertheless, we explicitly construct a counterexample. Further, we answer a question of D.Goss: is it possible that two Anderson t-motives that differ only by a nilpotent operator $N$ are of different uniformizability type, i.e. one of them is uniformizable and other not? We give an explicit example that this is possible.