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On Curves with Many Rational Points over Finite Fields

In: Finite Fields with Applications to Coding Theory, Cryptography and Related Areas

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  • Arnaldo Garcia

    (IMPA, Estrada Dona Castorina, 110)

Abstract

We summarize results on maximal curves over $${\mathbb{F}_{{q^2}}}$$ (i.e., curves attaining the Hasse-Weil upper bound for the number of rational points over finite fields). We discuss the classification problem and the genus spectrum of maximal curves. We present some towers of curves over finite fields attaining the Drinfeld-Vladut bound. Especially interesting is the description of the completely splitting locus (see Formula (20)) of a certain tower of curves, meaning the first description by their coordinates of the supersingular points of the modular curves X 0(2 n ), for each n ∈ ℕ.

Suggested Citation

  • Arnaldo Garcia, 2002. "On Curves with Many Rational Points over Finite Fields," Springer Books, in: Gary L. Mullen & Henning Stichtenoth & Horacio Tapia-Recillas (ed.), Finite Fields with Applications to Coding Theory, Cryptography and Related Areas, pages 152-163, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-59435-9_11
    DOI: 10.1007/978-3-642-59435-9_11
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