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2 results

  • On the non-existence of certain curves of genus two

  • Everett W. Howe
  • Journal:
    Compositio Mathematica / Volume 140 / Issue 3 / May 2004
    Published online by Cambridge University Press:
    04 December 2007, pp. 581-592
    Print publication:
    May 2004
    • Article
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  • We prove that if q is a power of an odd prime, then there is no genus-2 curve over $\mathbf{F}_q$ whose Jacobian has characteristic polynomial of Frobenius equal to $x^4 + (2 - 2q)x^2 + q^2$. Our proof uses the Brauer relations in a biquadratic extension of $\mathbb{Q}$ to show that every principally polarized abelian surface over $\mathbf{F}_q$ with the given characteristic polynomial splits over $\mathbf{F}_{q^2}$ as a product of polarized elliptic curves.