Non-Cyclic Subgroups of Jacobians of Genus Two Curves

2008/01/18 by Christian Robenhagen Ravnshøj, Christian Robenhagen Ravnshoj, Ravnshoj, Christian Robenhagen
Engineering · Mathematics · #11G20 (Primary) #11T71 #14G50 #14H45 (Secondary) #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Dynamics and Control of Mechanical Systems #FOS: Mathematics #math.AG #msc:11G20 #msc:11T71 #msc:14G50 #msc:14H45

paper · pdf · doi:10.48550/arxiv.0801.2835

arxiv created 2008/01/18 · openalex publication_date 2008/01/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let E be an elliptic curve defined over a finite field. Balasubramanian and Koblitz have proved that if the l-th roots of unity m_l is not contained in the ground field, then a field extension of the ground field contains m_l if and only if the l-torsion points of E are rational over the same field 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 field extension of the ground field. 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 field extension of degree at most 24.

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