Counting hyperelliptic curves

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We find a closed formula for the number hyp (g) of hyperelliptic curves of genus g over a finite field k = Fq of odd characteristic. These numbers hyp (g) are expressed as a polynomial in q with integer coefficients that depend on g and the set of divisors of q - 1 and q + 1. As a by-product we obtain a closed formula for the number of self-dual curves of genus g. A hyperelliptic curve is defined to be self-dual if it is k-isomorphic to its own hyperelliptic twist. © 2009 Elsevier Inc. All rights reserved.
Original languageEnglish
Pages (from-to)774-787
JournalAdvances in Mathematics
Publication statusPublished - 20 Jun 2009


  • Finite field
  • Hyperelliptic curve
  • Self-dual curve


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