On some bounds for symmetric tensor rank of multiplication in finite   fields
  We establish new upper bounds about symmetric bilinear complexity in any extension of finite fields. Note that these bounds are not asymptotical but uniform.   Moreover we give examples of Shimura curves that do not descend over their field of moduli, which discusses the validity of certain published bounds. 
St\'ephane Ballet, Julia Pieltant, Matthieu Rambaud and Jeroen   Sijsling
cs.IT math.IT