Dense families of modular curves, prime numbers and uniform symmetric tensor rank of multiplication in certain finite fields

摘要

We obtain new uniform bounds for the symmetric tensor rank of multiplicationin finite extensions of any finite field Fp or Fp2 where p denotes a primenumber greater or equal than 5. In this aim, we use the symmetricChudnovsky-type generalized algorithm applied on sufficiently dense families ofmodular curves defined over Fp2 attaining the Drinfeld-Vladuts bound and on thedescent of these families to the definition field Fp. These families areobtained thanks to prime number density theorems of type Hoheisel, inparticular a result due to Dudek (2016).