  We obtain new uniform upper bounds for the (non necessarily symmetric) tensor
rank of the multiplication in the extensions of the finite fields $\F_q$ for
any prime or prime power $q\geq2$; moreover these uniform bounds lead to new
asymptotic bounds as well. In addition, we also give purely asymptotic bounds
which are substantially better by using a family of Shimura curves defined over
$\F_q$, with an optimal ratio of $\F_{q^t}$-rational places to their genus
where $q^t$ is a square.
