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

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

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