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

Stéphane Ballet (1) , Alexey Zykin (2, 3, 4)

1 I2M - Institut de Mathématiques de Marseille
2 IITP - Institute for Information Transmission Problems
3 HSE - Vysšaja škola èkonomiki = National Research University Higher School of Economics [Moscow]
4 GAATI - Laboratoire de Géométrie Algébrique et Applications à la Théorie de l'Information

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 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 (Funct Approx Commmentarii Math, 55(2):177-197, 2016).

Keywords: Algebraic function field; Tower of function fields; Tensor rank; Algorithm; Finite field; Modular curve; Shimura curve

Domains: Number Theory [math.NT]

Stéphane Ballet, Alexey Zykin. Dense families of modular curves, prime numbers and uniform symmetric tensor rank of multiplication in certain finite fields. Designs, Codes and Cryptography, 2019, 87 (2-3), pp.517-525. ⟨10.1007/s10623-018-0560-8⟩. ⟨hal-02115440⟩