Shimura modular curves and asymptotic symmetric tensor rank of multiplication in any finite field

TL;DR

This paper establishes new asymptotic bounds for the symmetric tensor rank of multiplication in finite fields by leveraging Shimura modular curves and a generalized Chudnovsky algorithm, advancing understanding of finite field multiplication complexity.

Contribution

It introduces novel bounds using Shimura modular curves and a generalized algorithm, extending previous methods to all finite fields and their extensions.

Findings

New asymptotic bounds for symmetric tensor rank in finite fields

Application of Shimura modular curves attaining the Drinfeld-Vladut bound

Extension of bounds to all finite field extensions

Abstract

We obtain new asymptotical bounds for the symmetric tensor rank of multiplication in any finite extension of any finite field $\F_q$. In this aim, we use the symmetric Chudnovsky-type generalized algorithm applied on a family of Shimura modular curves defined over $\F_{q^2}$ attaining the Drinfeld-Vladut bound and on the descent of this family over the definition field $\F_q$.