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

## TL;DR

This paper establishes new uniform bounds for the symmetric tensor rank of multiplication in finite field extensions using dense families of modular curves and advanced prime number theorems.

## Contribution

It introduces a novel approach combining modular curves and prime density theorems to improve bounds on tensor rank in finite fields.

## Key findings

- New bounds for symmetric tensor rank in finite fields.
- Application of modular curves attaining the Drinfeld-Vladuts bound.
- Use of prime number density theorems like Hoheisel and Dudek.

## 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).