We obtain new uniform upper bounds for the (non necessarily symmetric) tensorrank of the multiplication in the extensions of the finite fields $\F_q$ forany prime or prime power $q\geq2$; moreover these uniform bounds lead to newasymptotic bounds as well. In addition, we also give purely asymptotic boundswhich 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 genuswhere $q^t$ is a square.
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