Title: Dense families of modular curves, prime numbers and uniform symmetric tensor rank of multiplication in certain finite fields,
Abstract: We obtain new uniform bounds for the symmetric tensor rank of multiplication
in finite extensions of any finite field Fp or Fp2 where p denotes a prime
number greater or equal than 5. In this aim, we use the symmetric
Chudnovsky-type generalized algorithm applied on sufficiently dense families of
modular curves defined over Fp2 attaining the Drinfeld-Vladuts bound and on the
descent of these families to the definition field Fp. These families are
obtained thanks to prime number density theorems of type Hoheisel, in
particular a result due to Dudek (2016).