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NON-CYCLIC SUBGROUPS OF JACOBIANS OF GENUS TWO CURVES

By Christian Robenhagen Ravnshøj

Abstract

Abstract. Let E be an elliptic curve de ned over a nite eld. Balasubramanian and Koblitz have proved that if the ℓth roots of unity µℓ is not contained in the ground eld, then a eld extension of the ground eld contains µℓ if and only if the ℓ-torsion points of E are rational over the same eld extension. We generalize this result to Jacobians of genus two curves. In particular, we show that the Weil- and the Tate-pairing are non-degenerate over the same eld extension of the ground eld. From this generalization we get a complete description of the ℓ-torsion subgroups of Jacobians of supersingular genus two curves. In particular, we show that for ℓ> 3, the ℓ-torsion points are rational over a eld extension of degree at most 24. 1

Year: 2012
OAI identifier: oai:CiteSeerX.psu:10.1.1.215.5913
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