On the nonexistence of certain curves of genus two

Details On the nonexistence of certain curves of genus two On the nonexistence of certain curves of genus two

2002-01-31

arxiv.org/abs/math/0201311v1

Compos. Math. 140 (2004) 581--592

Mathematics

Number Theory

LaTeX, 13 pages

Scientific paper

10.1112/S0010437X03000757

We prove that if q is a power of an odd prime then there is no genus-2 curve over 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 Q to show that every principally polarized abelian surface over F_q with the given characteristic polynomial splits over F_{q^2} as a product of polarized elliptic curves.

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