×

Symmetric Diophantine systems and families of elliptic curves of high rank. (English) Zbl 1457.11074

In this long paper the author, using an elementary approach to construct some elliptic curves over field of rational functions in several variables, with generic rank \(\geq 8\). The proposed method relies on the existence of rational solutions of certain symmetric Diophantine equations and is similar (but not the same) to the method used by J.-F. Mestre [C. R. Acad. Sci., Paris, Sér. I 313, No. 4, 171–174 (1991; Zbl 0749.14026)] to construct elliptic curves with high rank. The paper is easy to read and contains several examples.

MSC:

11G05 Elliptic curves over global fields
11D25 Cubic and quartic Diophantine equations
11D41 Higher degree equations; Fermat’s equation

Citations:

Zbl 0749.14026
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[2] B. Bektemirov, B. Mazur, W. Stein, and M. Watkins, Average ranks of elliptic curves: tension between data and conjecture, Bull. Amer. Math. Soc. (N.S.) \bf44 (2007), 233-254. · Zbl 1190.11032
[3] A. Choudhry and J. Wróblewski, Triads of integers with equal sums of squares and equal products and a related multigrade chain, Acta Arith., \bf178 (2017), 87-100. · Zbl 1428.11055
[4] J. W. S. Cassels, Diophantine equations with special reference to elliptic curves, J. London Math. Soc. 41 (1966), 193-291. · Zbl 0138.27002
[5] J. W. S. Cassels, Lectures on elliptic curves, Cambridge University Press (1991). · Zbl 0752.14033
[6] L. E. Dickson, Introduction to the theory of numbers, Dover, New York (1957). · Zbl 0084.26901
[7] S. Fermigier, Construction of high-rank elliptic curves over \(\mathbb{Q}\) and \(\mathbb{Q}(t)\) with non-trivial 2-torsion (extended abstract), pp.,in 115-120 Algorithmic number theory (Talence, 1996), Lecture Notes in Comput. Sci. \bf1122, Springer, Berlin (1996). · Zbl 0861.11035
[8] S. Kihara, On an elliptic curve over \(\mathbb{Q}(t)\) of rank \(\geq 14\), Proc. Japan Acad. Ser. A, \bf77 (2001), 50-51. · Zbl 1015.11022
[9] J.-F. Mestre, Courbes elliptiques de rang \(\ge11\) sur \(Q(t)\), C. R. Acad. Sci. Paris Sér. I Math. \bf313 (1991), 139-142. · Zbl 0745.14013
[10] J.-F. Mestre, Courbes elliptiques de rang \(\ge12\) sur \(Q(t)\), C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), 171-174. · Zbl 0749.14026
[11] L. J. Mordell, Diophantine equations, Academic Press, London (1969). · Zbl 0188.34503
[12] K. Nagao, An example of elliptic curve over \(\mathbb{Q}(T)\) with rank \(\geq 13\), Proc. Japan Acad. Ser. A, \bf70 (1994), 152-153. · Zbl 0848.14015
[13] J. Park, B. Poonen, J, Voight and M. M. Wood, A heuristic for boundedness of ranks of elliptic curves, arXiv:1602.01431 · Zbl 1469.11173
[14] B. Poonen, Heuristics for the arithmetic of elliptic curves, arXiv:1711.10112 · Zbl 1173.11040
[15] S. Schmitt and H. G. Zimmer, Elliptic curves: a computational approach, de Gruyter, Berlin (2003). · Zbl 1195.11078
[16] T. Shioda, Construction of elliptic curves over \(\mathbb{Q}(t)\) with high rank: a preview, Proc. Japan Acad. Ser. A, \bf66 (1990), 57-60. · Zbl 0762.14016
[17] J. H. Silverman, The arithmetic of elliptic curves, 2nd edition, Springer, Dordrecht (2009). · Zbl 1194.11005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.