New uniform and asymptotic upper bounds on the tensor rank of multiplication in extensions of finite fields

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.