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High rank elliptic curves with prescribed torsion group over quadratic fields

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Abstract

There are 26 possibilities for the torsion groups of elliptic curves defined over quadratic number fields. We present examples of high rank elliptic curves with a given torsion group which set the current rank records for most of the torsion groups. In particular, we show that for each possible torsion group, except maybe for \(\mathbb {Z}/15\mathbb {Z}\), there exists an elliptic curve over some quadratic field with this torsion group and with rank \(\ge 2\).

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Acknowledgments

The authors would like to thank Filip Najman and the referee for very useful comments on the previous version of this paper.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad del País Vasco, Aptdo. 644, 48080 , Bilbao, Spain

    Julián Aguirre & Juan Carlos Peral

  2. Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10000 , Zagreb, Croatia

    Andrej Dujella

  3. Department of Mathematics, University of Osijek, Trg Ljudevita Gaja 6, 31000 , Osijek, Croatia

    Mirela Jukić Bokun

Authors
  1. Julián Aguirre
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  2. Andrej Dujella
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  3. Mirela Jukić Bokun
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  4. Juan Carlos Peral
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Corresponding author

Correspondence to Andrej Dujella.

Additional information

A. D. was supported by the Ministry of Science, Education and Sports, Republic of Croatia, Grant 037-0372781-2821. J. C. P. was supported by the UPV/EHU Grant EHU 10/05.

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Aguirre, J., Dujella, A., Jukić Bokun, M. et al. High rank elliptic curves with prescribed torsion group over quadratic fields. Period Math Hung 68, 222–230 (2014). https://doi.org/10.1007/s10998-014-0032-5

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  • DOI: https://doi.org/10.1007/s10998-014-0032-5

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