  Let $\mathbf K$ be a finite field, $X$ and $Y$ two curves over $\mathbf K$,
and $Y\rightarrow X$ an unramified abelian cover with Galois group $G$. Let $D$
be a divisor on $X$ and $E$ its pullback on $Y$. Under mild conditions the
linear space associated with $E$ is a free ${\mathbf K}[G]$-module. We study
the algorithmic aspects and applications of these modules.
