Elliptic curve involving subfamilies of rank at least 5 over $\mathbb{Q}(t)$ or $\mathbb{Q}(t,k)$

Year 2021, , 721 - 731, 07.06.2021

Ahmed El Amine Youmbaı , A. Muhammed Uludağ , Djilali Behloul

https://doi.org/10.15672/hujms.708945

Cited By: 1

Abstract

Motivated by the work of Zargar and Zamani, we introduce a family of elliptic curves containing several one- (respectively two-) parameter subfamilies of high rank over the function field $\mathbb{Q}(t)$ (respectively $\mathbb{Q}(t,k)$). Following the approach of Moody, we construct two subfamilies of infinitely many elliptic curves of rank at least 5 over $\mathbb{Q}(t,k)$. Secondly, we deduce two other subfamilies of this family, induced by the edges of a rational cuboid containing five independent $\mathbb{Q}(t)$-rational points. Finally, we give a new subfamily induced by Diophantine triples with rank at least 5 over $\mathbb{Q}(t)$. By specialization, we obtain some specific examples of elliptic curves over $\mathbb{Q}$ with a high rank (8, 9, 10 and 11).

Keywords

Elliptic Curves, Rank, Rational Cuboid, Diophantine Triples

References

• [1] J. Aguirre, A. Dujella, M.J. Bokun and J.C. Peral, High rank elliptic curves with prescribed torsion group over quadratic fields, Period. Math. Hungar. 68 (2), 222– 230, 2014.
• [2] W. Bosma, J.J. Cannon, C. Fieker and A. Steel (eds.), Handbook of Magma Functions, Edition 2.20-9, 2014.
• [3] A. Brumer, The average rank of elliptic curves I, Invent. Math. 109 (1), 445–472, 1992.
• [4] G. Campbell, Finding elliptic curves and families of elliptic curves over $\mathbb{Q}$ of large rank, PhD Thesis, Rutgers University, 1999.
• [5] J.W.S. Cassels, Diophantine equations with special reference to elliptic curves, J. London. Math. Soc. 1 (1), 193-291, 1966.
• [6] L.E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine analysis, Dover Publications, New York, 2005.
• [7] A. Dujella, https://web.math.pmf.unizg.hr/~duje/index.html.
• [8] A. Dujella, On the size of Diophantine m-tuples, Math. Proc. Cambridge Philos. Soc. 132, 23–33, 2002.
• [9] A. Dujella, An example of elliptic curve over $\mathbb{Q}$ with rank equal to 15, Proc. Japan Acad. Ser. A Math. Sci. 78 (7), 109–111, 2002.
• [10] A. Dujella, On the number of Diophantine m-tuples, Ramanujan J. 15, 37–46, 2008.
• [11] A. Dujella and J.C. Peral, 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.
• [12] A. Dujella and V. Petricevic, Strong Diophantine triples, Exp. Math. 17, 83–89, 2008.
• [13] A. Filipin and A. Togbé, On the family of Diophantine triples $\left\lbrace k+2, 4k, 9k+6 \right\rbrace$, Acta Math. Acad. Paedagog. Nyhàzi. (N.S.)ü 25 (2), 145–153,2009.
• [14] B. He and A. Togbé, On the family of Diophantine triples $\left\lbrace k+1, 4k, 9k+3\right\rbrace $, Period. Math. Hungar. 58 (1), 59–70, 2009.
• [15] T. Honda, Isogenies, rational points and section points of group varieties, Jpn. J. Math. 30, 84–101, 1960.
• [16] J.F. Mestre, Construction de courbes elliptiques sur $\mathbb{Q}$ de rang $\geq 12$, C. R. Acad. Sci. Paris, Série I, 295, 643--644, 1982.
• [17] J.F. Mestre, Courbes elliptiques de rang $\geq 11$ sur $\mathbb{Q}(T)$, C. R. Acad. Sci. Paris, Série I, 313, 139-142, 1991.
• [18] D. Moody, M. Sadek and A.S. Zargar, Families of elliptic curves of rank $\geq 5$ over $\mathbb{Q}(t)$, Rocky Mountain J. Math. 49 (7), 2253–2266, 2019.
• [19] K. Nagao, An example of elliptic curve over $\mathbb{Q}$ with rank $> 20$, Proc. Japan Acad. Ser. A Math. Sci. 69, 291-293, 1993.
• [20] K. Nagao, An example of elliptic curve over $\mathbb{Q}$ with rank $\geq 21$, Proc. Japan Acad. Ser. A Math. Sci. 70, 104-105, 1994.
• [21] The PARI Group, PARI/GP version 2.9.1, Univ. Bordeaux, 2016, http://pari. math.u-bordeaux.fr/.
• [22] J. Park, B. Poonen, J. Voight and M.M. Wood, A heuristic for boundedness of ranks of elliptic curves, J. Eur. Math. Soc. 21, 2859–2903, 2019, doi: 10.4171/JEMS/893.
• [23] N. Saunderson, The Elements of Algebra, Book 6, Cambridge University Press, Cambridge, 1740.
• [24] J.H. Silverman, The Arithmetic of Elliptic Curves, 2nd Edition, Vol. 106, Springer Science and Business Media, 2009.
• [25] J.H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Vol. 151, Springer Science and Business Media, 2013.
• [26] J.T. Tate, The arithmetic of elliptic curves, Invent. Math. 23 (3-4), 179–206, 1974.
• [27] A.S. Zargar and N. Zamani, A Family of Elliptic Curves of Rank $\geq 5$ over $\mathbb{Q}(m)$, Notes Number Theory Discrete Math. 25 (4), 24–29, 2019.