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Cube sums of the forms 3p and \(3p^2\)

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Abstract

Let \(p\equiv 2,5\ \mathrm {mod}\ 9\) be a prime. In this paper, we prove that at least one of 3p and \(3p^2\) is a cube sum by constructing certain nontrivial Heegner points. We also establish the explicit Gross–Zagier formulae for these Heegner points and give variants of the Birch and Swinnerton–Dyer conjecture of the related elliptic curves.

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Acknowledgements

The authors would like to thank professor Ye Tian for his long-term support and encouragement. The second-named author would like to thank professor Chen-Bo Zhu for his encouragement and support. We would like to thank John Voight for his suggestion to use the Kronecker congruence to compute the coordinates modulo p in the supersingular case. We also would like to thank Jinbang Yang for helpful discussions which lead a proof of Proposition 3.5, and Sheng Meng for valuable discussions on algebraic geometry. We thank the referee for the careful reading and the helpful advices. All the numerical computations are programed on the SageMath system.

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Authors and Affiliations

  1. School of Mathematical Sciences, Tongji University, Shanghai, 200092, China

    Jie Shu

  2. Academy of Mathematics and System Sciences, The Chinese Academy of Sciences, Beijing, 100190, China

    Xu Song

  3. School of Mathematics, Shandong University, Jinan, Shandong, 250100, China

    Hongbo Yin

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  1. Jie Shu
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  2. Xu Song
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  3. Hongbo Yin
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Correspondence to Hongbo Yin.

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Jie Shu is supported by NSFC-11701092; Xu Song is supported by the International Postdoctoral Exchange Fellowship Program (Talent-Introduction Program) of the OCPC. Hongbo Yin is supported by NSFC-11701548 and The Young Scholars Program of Shandong University.

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Shu, J., Song, X. & Yin, H. Cube sums of the forms 3p and \(3p^2\). Math. Z. 299, 2297–2325 (2021). https://doi.org/10.1007/s00209-021-02730-w

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