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Rational torsion of J0(N) for hyperelliptic modular curves and families of Jacobians of genus 2 and genus 3 curves with a rational point of order 5,7 or 10

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Abstract

We describe a way of constructing Jacobians of hyperelliptic curves of genus g ≥ 2, defined over a number field, whose Jacobians have a rational point of order of some (well chosen) integer l ≥ g + 1; the method is based on a polynomial identity. Using this approach we construct new families of genus 2 curves defined over — which contain the modular curves X0(31) (and X0(22) as a by-product) and X0(29), the Jacobians of which have a rational point of order 5 and 7 respectively. We also construct a new family of hyperelliptic genus 3 curves defined over —, which contains the modular curve X0(41), the Jacobians of which have a rational point of order 10. Finally we show that all hyperelliptic modular curves X0(N) with N a prime number fit into the described strategy, except for N = 37 in which case we give another explanation.

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Authors and Affiliations

  1. LIASIT, Université du Luxembourg, 162 A, Avenue de la Faïencerie, 1511, Luxembourg

    F. Leprévost

  2. Fakultät II - Mathematik MA 8-1, Technische Universität Berlin, Straβe des 17, Juni 136, 10623, Berlin, Germany

    M. Pohst & A. Schöpp

Authors
  1. F. Leprévost
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  2. M. Pohst
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  3. A. Schöpp
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Corresponding authors

Correspondence to F. Leprévost, M. Pohst or A. Schöpp.

Additional information

R. Berndt

The authors thank the FNR (project FNR/04/MA6/11) for their support.

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Leprévost, F., Pohst, M. & Schöpp, A. Rational torsion of J0(N) for hyperelliptic modular curves and families of Jacobians of genus 2 and genus 3 curves with a rational point of order 5,7 or 10. Abh.Math.Semin.Univ.Hambg. 74, 193–203 (2004). https://doi.org/10.1007/BF02941535

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