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The Mordell-Weil bases for the elliptic curve y2 = x3 − m2x + m2

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Abstract

Let Dm be an elliptic curve over ℚ of the form y2 = x3 − m2x + m2, where m is an integer. In this paper we prove that the two points P−1 = (−m, m) and P0 = (0, m) on Dm can be extended to a basis for Dm(ℚ) under certain conditions described explicitly.

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Acknowledgment

The authors sincerely thank to Prof. Y. Fujita for his useful suggestions during the preparation of this manuscript. Also, the authors thank to the referee for many valuable suggestions and comments which improved the readability of this paper. This work was started when the second author visited the Institute of Mathematics & Applications Bhubaneswar. He thanks the people of this institute for the hospitality and support. The second author also thanks to the IMSc Chennai for providing research facilities to pursue his research work.

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Authors and Affiliations

  1. Institute of Mathematics & Applications, Andharua, Bhubaneswar, 751029, India

    Sudhansu Sekhar Rout

  2. Institute of Mathematical Sciences (HBNI), CIT Campus, Taramani, Chennai, 600 113, India

    Abhishek Juyal

Authors
  1. Sudhansu Sekhar Rout
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  2. Abhishek Juyal
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Corresponding author

Correspondence to Abhishek Juyal.

Additional information

A. Juyal’s research was supported by IMSc Chennai (HBNI) Post-Doctoral Fellowship.

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Rout, S.S., Juyal, A. The Mordell-Weil bases for the elliptic curve y2 = x3 − m2x + m2. Czech Math J 71, 1133–1147 (2021). https://doi.org/10.21136/CMJ.2021.0238-20

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  • DOI: https://doi.org/10.21136/CMJ.2021.0238-20

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