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

Stéphane Ballet (1) , Jean Chaumine (2) , Julia Pieltant (3, 4)

1 IML - Institut de mathématiques de Luminy
2 GAATI - Laboratoire de Géométrie Algébrique et Applications à la Théorie de l'Information
3 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
4 GRACE - Geometry, arithmetic, algorithms, codes and encryption

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-Vl\u{a}du\c{t} bound and on the descent of this family over the definition field $\F_q$.

Mots clés: Algebraic function field; tower of function fields; tensor rank; algorithm; finite field; modular curve; Shimura curve

Domaines: Géométrie algébrique [math.AG]; Théorie des nombres [math.NT]; Complexité [cs.CC]

Stéphane Ballet, Jean Chaumine, Julia Pieltant. Shimura modular curves and asymptotic symmetric tensor rank of multiplication in any finite field. Conference on Algebraic Informatics, Sep 2013, Porquerolles Island, France. pp.160-172, ⟨10.1007/978-3-642-40663-8_16⟩. ⟨hal-00828070⟩