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We obtain new asymptotical bounds for the symmetric tensor rank ofmultiplication in any finite extension of any finite field $\F_q$. In this aim,we use the symmetric Chudnovsky-type generalized algorithm applied on a familyof Shimura modular curves defined over $\F_{q^2}$ attaining the Drinfeld-Vladutbound and on the descent of this family over the definition field $\F_q$.

Shimura modular curves and asymptotic symmetric tensor rank of multiplication in any finite field