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

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We obtain new uniform upper bounds for the (non necessarily symmetric) tensor rank of the multiplication in the extensions of the finite fields for any prime or prime power q≥2; 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, with an optimal ratio of ₜ-rational places to their genus where qᵗ is a square.