On the nonexistence of certain curves of genus two
Abstract
We prove that if q is a power of an odd prime then there is no genus-2 curve over Fq whose Jacobian has characteristic polynomial of Frobenius equal to x4 + (2-2q)x2 + q2. Our proof uses the Brauer relations in a biquadratic extension of Q to show that every principally polarized abelian surface over Fq with the given characteristic polynomial splits over Fq2 as a product of polarized elliptic curves.
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