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Supersingular genus-two curves over fields of characteristic three

Let C be a supersingular genus-2 curve over an algebraically closed field of characteristic 3. We show that if C is not isomorphic to the curve y^2 = x^5 + 1 then up to isomorphism there are exactly 20 degree-3 maps phi from C to the elliptic curve E with j-invariant 0. We study the coarse moduli space of triples (C,E,phi), paying particular attention to questions of rationality. The results we obtain allow us to determine, for every finite field k of characteristic 3, the polynomials that occur as Weil polynomials of supersingular genus-2 curves over k.