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Construction of High Rank Elliptic Curves

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Abstract

We list a number of strategies for construction of elliptic curves having high rank with special emphasis on those curves induced by Diophantine triples, in which we have contributed more. These strategies have been developed by many authors. In particular we present a new example of a curve, induced by a Diophantine triple, with torsion \(\mathbb {Z}/ 2 \mathbb {Z}\times \mathbb {Z}/ 4\mathbb {Z}\) and with rank 9 over \(\mathbb {Q}\). This is the present record for this kind of curves.

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References

  1. Aguirre, J., Castaneda, F., Peral, J.C.: High rank elliptic curves with torsion group \({\mathbb{Z}}/(2{\mathbb{Z}} )\). Math. Comput. 73(245), 323–331 (2003)

    Article  Google Scholar 

  2. Aguirre, J., Dujella, A., Peral, J.C.: On the rank of elliptic curves coming from rational Diophantine triples. Rocky Mountain J. Math. 42, 1759–1776 (2012)

    Article  MathSciNet  Google Scholar 

  3. Campbell, G., Goins, E. H.: Heron triangles, Diophantine problems and elliptic curves, preprint

  4. Carmichael, R.D.: Diophantine Analysis. Dover, New York (1959)

    MATH  Google Scholar 

  5. Cremona, J.: Algorithms for Modular Elliptic Curves. Cambridge University Press, Cambridge (1997)

    MATH  Google Scholar 

  6. Dickson, L.: History of the Theory of Numbers. Chelsea, New York (1971)

    Google Scholar 

  7. Dujella, A.: High rank elliptic curves with prescribed torsion, http://web.math.hr/~duje/tors/tors.html

  8. Dujella, A.: Some polynomial formulas for Diophantine quadruples. Grazer Math. Ber. 328, 25–30 (1996)

    MathSciNet  MATH  Google Scholar 

  9. Dujella, A.: Diophantine \(m\)-tuples and elliptic curves. J. Théor. Nr. Bordx. 13, 111–124 (2001)

    Article  MathSciNet  Google Scholar 

  10. Dujella, A.: On Mordell–Weil groups of elliptic curves induced by Diophantine triples. Glas. Mat. Ser. III(42), 3–18 (2007)

    Article  MathSciNet  Google Scholar 

  11. Dujella, A., Peral, J.C.: High rank elliptic curves with torsion \({\mathbb{Z}}/2{\mathbb{Z}} \times {\mathbb{Z}}/4{\mathbb{Z}}\) induced by Diophantine triples. LMS J. Comput. Math. 17, 282–288 (2014)

    Article  MathSciNet  Google Scholar 

  12. Dujella, A., Peral, J.C.: Elliptic curves with torsion group \({\mathbb{Z}}/8{\mathbb{Z}} \) or \({\mathbb{Z}}/2{\mathbb{Z}} \times {\mathbb{Z}}/6{\mathbb{Z}}\). Trends Number Theory Contemp. Math. 649, 47–62 (2015)

    Article  Google Scholar 

  13. Dujella, A., Peral, J.C.: Elliptic curves induced by Diophantine triples. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM 113, 791–806 (2019)

    Article  MathSciNet  Google Scholar 

  14. Elkies, N.: Three lectures on elliptic surfaces and curves of high rank, arXiv:0709.2908v1 (2007)

  15. Elkies, N.D.: \(E({\mathbb{Q}}) = ({\mathbb{Z}}/2{{\mathbb{Z}}}) \times ({{\mathbb{Z}}}/4{{\mathbb{Z}}}) \times {{\mathbb{Z}}}^8\). Number Theory Listserver. Cambridge University Press, Cambridge (2005)

    Google Scholar 

  16. Eroshkin, Y. G.: Personal communication, (2008)

  17. Fermigier, S.: Construction of high rank elliptic curves over \({{\mathbb{Q}}}\) and over \({{\mathbb{Q}}}(t)\) with nontrivial \(2\)-torsion. In: Algorithmic Number Theory (Talence 1996), Springer, Berlin (1996)

  18. Gusić, I., Tadić, P.: A remark on the injectivity of the specialization homomorphism. Glas. Mat. Ser. III(47), 265–275 (2012)

    Article  MathSciNet  Google Scholar 

  19. Gusić, I., Tadić, P.: Injectivity of the specialization homomorphism of elliptic curves. J. Number Theory 148, 137–152 (2015)

    Article  MathSciNet  Google Scholar 

  20. Kubert, S.: Universal bounds on the torsion of elliptic curves. Proc. Lond. Math. Soc. 3(33), 193–237 (1976)

    Article  MathSciNet  Google Scholar 

  21. Kamienny, S.: Torsion on elliptic curves and q-coefficients of modular forms. Invent. Math. 109(2), 221–229 (1992)

    Article  MathSciNet  Google Scholar 

  22. Klagsbrun, Z., Sherman, T., Weigandt, J.: The Elkies curve has rank \(28\) subject only to GRH. Math. Comput. 88, 837–846 (2019)

    Article  MathSciNet  Google Scholar 

  23. Kihara, S.: On the rank of elliptic curves with a rational point of order \(4\). Proc. Jpn. Acad. Ser. A 80, 26–27 (2004)

    MathSciNet  MATH  Google Scholar 

  24. Kihara, S.: On the rank of elliptic curves with a rational point of order \(4\), II. Proc. Jpn. Acad. Ser. A 80, 158–159 (2004)

    MathSciNet  MATH  Google Scholar 

  25. Kenku, M.A., Momose, F.: Torsion points on elliptic curves defined over quadratic fields. Nagoya Math. J. 109, 125–149 (1988)

    Article  MathSciNet  Google Scholar 

  26. Knapp, A.: Elliptic Curves. Princeton University Press, Princeton (1992)

    MATH  Google Scholar 

  27. Khoshnam, F., Moody, D.: High rank elliptic curves with torsion \( {\mathbb{Z}}/4{{\mathbb{Z}}}) \) induced by Kiharaś Elliptic curves. Integers 16, 1–13 (2016)

    MathSciNet  Google Scholar 

  28. Kulesz, L.: Families of elliptic curves of high rank with nontrivial torsion group over \({\mathbb{Q}}\). Acta Arith. 108, 339–356 (2003)

    Article  MathSciNet  Google Scholar 

  29. Lecacheux, O.: Rang de courbes elliptiques dont le groupe de torsion est non trivial. Ann. Sci. Math. Que. 28, 145–151 (2004)

    MathSciNet  MATH  Google Scholar 

  30. MacLeod, A.: A simple method for high-rank families of elliptic curves with specified torsion, arXiv:1410.1662v1 [math. NT] (2014)

  31. Mestre, J.F.: Courbes elliptiques de rang \(\ge 11\) sur \({\mathbb{Q}}(t)\). C. R. Acad. Sci. Paris. Sér. Math. 313, 139–142 (1991)

    MathSciNet  MATH  Google Scholar 

  32. Mestre, J.F.: Rang de courbes elliptiques d’invariant donné. C. R. Acad. Sci. Paris Sér. Math. 314, 919–922 (1992)

    MathSciNet  MATH  Google Scholar 

  33. Mazur, B.: Modular curves and Eisenstein ideal. Inst. Ht. Études Sci. Publ. Math. 47, 33–186 (1977)

    Article  MathSciNet  Google Scholar 

  34. Mordell, L.J.: On the rational solutions of the indeterminate equations of the third and fourth degree. Proc. Camb. Phil. Soc. 21, 179–192 (1922)

    MATH  Google Scholar 

  35. Park, J., Poonen, B., Voight, J., Wood, M.M.: A heuristic for boundedness of ranks of elliptic curves. J. Eur. Math. Soc. (JEMS) 21, 2859–2903 (2019)

    Article  MathSciNet  Google Scholar 

  36. Rabarison, P.: Structure de torsion des courbes elliptiques definies sur les corps de nombres quadratiques. Acta Arith. 144, 17–52 (2010)

    Article  MathSciNet  Google Scholar 

  37. Shioda, T.: An infinite family of elliptic curves over \({\mathbb{Q}}\) with large rank via Neron’s method. Invent. Math. 106, 109–119 (1991)

    Article  MathSciNet  Google Scholar 

  38. Silverman, J.H.: Advanced Topics in the Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, vol. 151. Springer, New York (1994)

    Book  Google Scholar 

  39. Weil, A.: L’arithmétique sur les courbes algébriques. Acta Math. 52, 281–315 (1929)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors want to deeply thank to the anonymous referees for a careful reading of the paper and for valuable suggestions which improved the presentation.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička cesta 30, 10000, Zagreb, Croatia

    Andrej Dujella

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

    Juan Carlos Peral

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  1. Andrej Dujella
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  2. Juan Carlos Peral
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Correspondence to Juan Carlos Peral.

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To the memory of Elias M. Stein with admiration and gratitude.

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A.D. acknowledges support from the QuantiXLie Center of Excellence, a project co-financed by the Croatian Government and European Union through the European Regional Development Fund—the Competitiveness and Cohesion Operational Programme (Grant KK.01.1.1.01.0004). A.D. was also supported by the Croatian Science Foundation under the Project No. IP-2018-01-1313. J.C.P. was supported by the Grant MTM2014-52347-C2-1-R.

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Dujella, A., Peral, J.C. Construction of High Rank Elliptic Curves. J Geom Anal 31, 6698–6724 (2021). https://doi.org/10.1007/s12220-020-00373-7

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  • DOI: https://doi.org/10.1007/s12220-020-00373-7

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