Abstract
In this paper, we present details of seven elliptic curves over \({\mathbb {Q}}(u)\) with rank 2 and torsion group \({\mathbb {Z}}/8{\mathbb {Z}}\) and five curves over \({\mathbb {Q}}(u)\) with rank 2 and torsion group \({\mathbb {Z}}/2{\mathbb {Z}}\times {\mathbb {Z}}/6{\mathbb {Z}}\). We also exhibit some particular examples of curves with high rank over \({\mathbb {Q}}\) by specialization of the parameter. We present several sets of infinitely many elliptic curves in both torsion groups and rank at least 3 parametrized by elliptic curves having positive rank. In some of these sets we have performed calculations about the distribution of the root number. This has relation with recent heuristics concerning the rank bound for elliptic curves by Park, Poonen, Voight and Wood.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beauville, A.: Les familles stables de courbes elliptiques sur $P^1$ admettant quatre fibres singulières. C. R. Acad. Sci. Paris Sér. I Math. 294, 657–660 (1982)
Bertin, M.J., Lecacheux, O.: Elliptic fibrations on the modular surface associated to $\Gamma _1(8)$. In: Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds, Fields Institute Communications, vol. 67, pp. 153–199. Springer, New York (2013)
Campbell, G.: Finding elliptic curves defined over ${\mathbb{Q}}$ of high rank. In: African Americans in Mathematics (Piscataway, NJ, 1996). DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 34, pp. 107–109. American Mathematical Society, Providence (1997)
Cremona, J.: Algorithms for Modular Elliptic Curves. Cambridge University Press, Cambridge (1997)
Dujella, A.: High rank elliptic curves with prescribed torsion. http://web.math.hr/~duje/tors/tors.html
Dujella, A., Bokun, M.J., Soldo, I.: On the torsion group of elliptic curves induced by Diophantine triples over quadratic fields. Rev. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 111(4), 1177–1185 (2017)
Dujella, A.: On Mordell–Weil groups of elliptic curves induced by Diophantine triples. Glas. Mat. Ser. III(42), 3–18 (2007)
Dujella, A., Peral, J.C.: Elliptic curves with torsion group ${\mathbb{Z}} / 8 {\mathbb{Z}}$ or ${\mathbb{Z}} /2 {\mathbb{Z}} \times {\mathbb{Z}} / 6 {\mathbb{Z}}$ (2013). arXiv:1306.0027
Dujella, A., Peral, J.C.: Elliptic curves with torsion group ${\mathbb{Z}} / 8 {\mathbb{Z}}$ or ${\mathbb{Z}} / 2 {\mathbb{Z}} \times {\mathbb{Z}} / 6 {\mathbb{Z}}$. In: Trends in Number Theory. Contemporary Mathematics, vol. 649, pp. 47–62 (2015)
Dujella, A., Peral, J.C.: Elliptic curves induced by Diophantine triples. Rev. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 113(2), 791–806 (2019)
Elkies, N. D.: Some more rank records: $E(\mathbb{Q}) = (\mathbb{Z}/2\mathbb{Z}) \times \mathbb{Z}^{18}, (\mathbb{Z}/4\mathbb{Z}) \times \mathbb{Z}^{12}, (\mathbb{Z}/8\mathbb{Z}) \times \mathbb{Z}^{6}, (\mathbb{Z}/2\mathbb{Z}) \times (\mathbb{Z}/6\mathbb{Z}) \times \mathbb{Z}^{6}$, Number Theory Listserver (2006)
Gusić, I., Tadić, P.: Injectivity of the specialization homomorphism of elliptic curves. J. Number Theory 148, 137–152 (2015)
Knapp, A.: Elliptic Curves. Princeton University Press, Princeton (1992)
Kulesz, L.: Families of elliptic curves of high rank with nontrivial torsion group over ${\mathbb{Q}}$. Acta Arith. 108, 339–356 (2003)
Lecacheux, O.: Rang de courbes elliptiques. Acta Arith. 109, 131–142 (2003)
Lecacheux, O.: Rang de courbes elliptiques dont le groupe de torsion est non trivial. Ann. Sci. Math. Québec 28, 145–151 (2004)
Livné, R., Yui, N.: The modularity of certain non-rigid Calabi–Yau threefolds. J. Math. Kyoto Univ. 45, 645–665 (2005)
MacLeod, A.J.: A simple method for high-rank families of elliptic curves with specific torsion (2014). arXiv:1410.1662
Park, J., Poonen, B., Voight, J., Wood, M.M.: A heuristic for boundedness of ranks of elliptic curves. J. Eur. Math. Soc. 21, 2859–2903 (2019)
Rabarison, P.: Construction of infinite family of elliptic curves with non trivial torsion group and high rank. Internal report 2009-17-LMNO-CNRS UMR 6139 (2009)
Silverman, J.H.: Advanced Topics in the Arithmetic of Elliptic Curves. Springer, New York (1994)
Voznyy, M.: Personal communication (2021)
Woo, J.: Arithmetic of elliptic curves and surfaces. Descent and quadratic sections. Doctoral Dissertation, Harvard University (2010)
Acknowledgements
The authors would like to thank Ivica Gusić and Maksym Voznyy for useful comments on the previous version of this paper, and to the members of Mersenne Forum (https://mersenneforum.org/) for the help with factorization in Sect. 11. The authors also would like to thank to the anonymous referees for several helpful suggestion.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
A.D. and M.K. were supported by the Croatian Science Foundation under the Project no. IP-2018-01-1313 and 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). J.C.P. was supported by the UPV/EHU Grant EHU 10/05. J.C.P. dedicate this paper to Ireneo Peral in memoriam.
Rights and permissions
About this article
Cite this article
Dujella, A., Kazalicki, M. & Peral, J.C. Elliptic curves with torsion groups \({ \mathbb {Z}}/ 8 {\mathbb {Z} } \) and \({ \mathbb {Z}}/ 2 {\mathbb {Z}} \times {\mathbb {Z}}/ 6 {\mathbb {Z}}\). Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 115, 169 (2021). https://doi.org/10.1007/s13398-021-01112-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13398-021-01112-5