Skip to main content
SpringerLink
Log in

Construction of an infinite family of elliptic curves of 2-Selmer rank 1 from Heron triangles

  • Research
  • Published:
Research in Number Theory Aims and scope Submit manuscript

Abstract

Given any positive integer n, there exist triangles, called Heron triangles, with rational sides whose area is n. Assuming the finiteness of the Shafarevich–Tate group, we construct a family of infinitely many Heronian elliptic curves of rank 1 arising from Heron triangles of a certain type. We then explicitly produce a separate family of infinitely many Heronian elliptic curves with 2-Selmer rank lying between 1 and 3.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data Availability

We confirm all the data with their respective sources are included and properly cited in the article.

References

  1. Birch, B.J., Swinnerton-Dyer, H.P.F.: Notes on elliptic curves II. J. Reine Angew. Math. 218, 79–108 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  2. Heath-Brown, D.R.: The size of Selmer groups for the congruent number problem. Invent. Math. 111(1), 171–195 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  3. Heath-Brown, D.R.: The size of Selmer groups for the congruent number problem, II. Invent. Math. 118, 331–370 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  4. Buchholz, R.H., Randall, L.R.: An infinite set of Heron triangles with two rational medians. Am. Math. Mthly. 104(2), 107–115 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Buchholz, R.H., Stingley, R.P.: Heron triangles with three rational medians. Rocky Mt. J. Math. 49(2), 405–417 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cassels, J.W.S.: Arithmetic on Curves of Genus. 1. IV. Proof of the Hauptvermutung. J. Reine Angew. Math. 211, 95–112 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chahal, J.S.: Congruent numbers and elliptic curves. Am. Math. Mthly. 113(4), 308–317 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Coates, J.H.: Congruent number problem. Pure Appl. Math. Q. 1(1), 14–27 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Connell, I.: Calculating root numbers of elliptic curves over Q. Manuscr. Math. 82(1), 93–104 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  10. Conrad, K.: The congruent number problem. Harvard College Math. Rev. 2(2), 58–74 (2008)

    Google Scholar 

  11. Dujella, A., Peral, J.C.: Elliptic curves coming from Heron triangles. Rocky Mt. J. Math. 44(4), 1145–1160 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ghale, V., Das, S., Chakraborty, D.: A Heron triangle and a Diophantine equation. Period. Math. Hung 1, 1–8 (2022)

    Google Scholar 

  13. Goins, E.H., Maddox, D.: Heron triangles via elliptic curves. Rocky Mt. J. Math. 1, 1511–1526 (2006)

    MathSciNet  MATH  Google Scholar 

  14. Halbeisen, L., Norbert, H.: Heron triangles and their elliptic curves. J. Number Theory 213, 232–253 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  15. Johnstone, J.A., Spearman, B.K.: Congruent number elliptic curves with rank at least three. Can. Math. Bull. 53(4), 661–666 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Koblitz, N.I.: Introduction to Elliptic Curves and Modular Forms, vol. 97. Springer, Berlin (2012)

    MATH  Google Scholar 

  17. Kramer, A.V., Luca, F.: Some remarks on Heron triangles. Acta Acad. Paed. Agriensis Sect. Math. 27, 25–38 (2000)

    MathSciNet  MATH  Google Scholar 

  18. The MAGMA algebra system. http://magma.maths.usyd.edu.au/calc/

  19. Rhoades, R.C.: 2-Selmer groups and the Birch–Swinnerton–Dyer Conjecture for the congruent number curves. J. Number Theory 129(6), 1379–1391 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Rusin, D.J.: Rational triangles with equal area. N. Y. J. Math. 4(1), 15 (1998)

    MathSciNet  MATH  Google Scholar 

  21. Sierpinski, W.: Pythagorean Triangles, vol. 9. Courier Corporation, Chelmsford (2003)

    MATH  Google Scholar 

  22. Silverman, J.H.: The Arithmetic of Elliptic Curves, vol. 106. Springer, New York (2009)

    MATH  Google Scholar 

  23. Stein, W.: Sage: Open source mathematical software. http://www.sagemath.org

  24. Washington, L.C.: Elliptic Curves: Number Theory and Cryptography. Chapman and Hall/CRC, New York (2008)

    Book  MATH  Google Scholar 

  25. Yan, X.: The Diophantine equation \((m^2+ n^2)^x+(2mn)^y=(m+ n)^{2z}\). Int. J. Number Theory 16(08), 1701–1708 (2020)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The first author would like to acknowledge DST-SERB for providing the grant through the Start-Up Research Grant (SRG/2020/001937) as well as BITS-Pilani, Hyderabad Campus for providing amenities. The second author would like to acknowledge the fellowship (File No: 09/1026(0029)/2019-EMR-I) and amenities provided by the Council of Scientific and Industrial Research, India (CSIR) and BITS-Pilani, Hyderabad. The third author would like to thank MATRICS, SERB for their research Grant MTR/2020/000467. The authors would also like to thank the anonymous referees for going through the manuscript very carefully and making valuable suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, BITS-Pilani, Hyderabad Campus, Hyderabad, India

    Debopam Chakraborty & Vinodkumar Ghale

  2. Department of Mathematics and Computation, IIT Guwahati, Guwahati, India

    Anupam Saikia

Authors
  1. Debopam Chakraborty
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vinodkumar Ghale
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Anupam Saikia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Debopam Chakraborty.

Ethics declarations

Conflict of interest

All authors declare that they have no conflicts of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chakraborty, D., Ghale, V. & Saikia, A. Construction of an infinite family of elliptic curves of 2-Selmer rank 1 from Heron triangles. Res. number theory 8, 101 (2022). https://doi.org/10.1007/s40993-022-00411-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40993-022-00411-z

Keywords

Mathematics Subject Classification

Search

Navigation