On the tensor rank of the multiplication in the finite fields Original Research Article

First, we prove the existence of certain types of non-special divisors of degree g−1 in the algebraic function fields of genus g defined over image. Then, it enables us to obtain upper bounds of the tensor rank of the multiplication in any extension of quadratic finite fields image by using Shimura and modular curves defined over image. From the preceding results, we obtain upper bounds of the tensor rank of the multiplication in any extension of certain non-quadratic finite fields image, notably in the case of image. These upper bounds attain the best asymptotic upper bounds of Shparlinski–Tsfasman–Vladut [I.E. Shparlinski, M.A. Tsfasman, S.G. Vladut, Curves with many points and multiplication in finite fields, in: Lecture Notes in Math., vol. 1518, Springer-Verlag, Berlin, 1992, pp. 145–169].