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