Elliptic curves of high rank with nontrivial torsion group over \(\mathbb{Q}\). (English) Zbl 1060.11036

Kulesz, Leopoldo; Stahlke, Colin

Exp. Math. 10, No. 3, 475-480 (2001).

Authors’ introduction: In order to find elliptic curves over \(\mathbb Q\) with large rank, J.-F. Mestre [C. R. Acad. Sci., Paris, Sér. I 313, No. 4, 171–174 (1991; Zbl 0749.14026)] constructed an infinite family of elliptic curves with rank at least 12. Then K. Nagao [Proc. Japan Acad., Ser. A 70, No. 5, 152–153 (1994; Zbl 0848.14015)] and S. Kihara [Proc. Japan Acad., Ser. A 73, No. 2, 32 (1997; Zbl 0906.11023)] found infinite subfamilies of rank 13 and 14, respectively. By specialization, elliptic curves of rank at least 21 (Nagao and Kouya, 1994), 22 (Fermigier, 1997), and 23 (Martin and McMillen, 1997) have also been found.
There have been quite a few efforts to construct families of curves with large rank having a nontrivial prescribed torsion group [cf. S. Fermigier, C. R. Acad. Sci., Paris, Sér. I 322, No. 10, 949–957 (1996; Zbl 0861.11035), and also Nagao, Kihara, Kulesz]. For example, Fermigier constructed a family of elliptic curves with rank at least 8 and a nontrivial point of order 2 and specialized it to a curve of rank 14.
Here we improve the rank records for curves with torsion group \(\mathbb Z/3\mathbb Z, \mathbb Z/4\mathbb Z, \mathbb Z/5\mathbb Z, \mathbb Z/6\mathbb Z, \mathbb Z/7\mathbb Z\), \(\mathbb Z/8\mathbb Z\), and \(\mathbb Z/2\mathbb Z\times \mathbb Z/2\mathbb Z\).

MSC:

11G05

Elliptic curves over global fields

Keywords:

high Mordell-Weil rank

Citations:

Zbl 0749.14026; Zbl 0848.14015; Zbl 0906.11023; Zbl 0861.11035

References:

[1]	Fermigier S., C. R. Acad. Sci. Paris Sér. I Math. 322 (10) pp 949– (1996)
[2]	Fermigier S., Acta Arith. 82 (4) pp 359– (1997)
[3]	Kihara S., Proc. Japan Acad. Ser. A Math. Sci. 73 (9) pp 165– (1997) · Zbl 0906.11026 · doi:10.3792/pjaa.73.165
[4]	Kihara S., Proc. Japan Acad. Ser. A Math. Sci. 73 (2) pp 32– (1997) · Zbl 0906.11023 · doi:10.3792/pjaa.73.32
[5]	Kihara S., Proc. Japan Acad. Ser. A Math. Sci. 73 (8) pp 151– (1997) · Zbl 0906.11025 · doi:10.3792/pjaa.73.151
[6]	Knapp A., Elliptic curves (1992) · Zbl 0804.14013
[7]	Kulesz L., Ph.D. thesis, in: Arithmétique des courbes algébriques de genre au moins deux (1998)
[8]	Kulesz L., C. R. Acad. Sci. Paris Sér. I Math. 329 (6) pp 503– (1999)
[9]	Martin, R. and McMillen, W. 1997. [Martin and McMillen 1997],http://www.math.niu.edu/ rusin/known-math/98/hirankinformal report, See
[10]	Mestre J.-F., C. R. Acad. Sci. Paris Sér. I Math. 313 (4) pp 171– (1991)
[11]	Mordell L. J., Diophantine equations (1969) · Zbl 0188.34503
[12]	Nagao K., Proc. Japan Acad. Ser. A Math. Sci. 70 (5) pp 152– (1994) · Zbl 0848.14015 · doi:10.3792/pjaa.70.152
[13]	Nagao K., Math. Comp. 66 (217) pp 411– (1997) · Zbl 0854.11029 · doi:10.1090/S0025-5718-97-00779-5
[14]	Nagao K., Pro Japan Acad. Ser. A Math. Sci. 70 (4) pp 104– (1994) · Zbl 0832.14022 · doi:10.3792/pjaa.70.104
[15]	Stoll M., Acta Arith. 98 pp 245– (2001) · Zbl 0972.11058 · doi:10.4064/aa98-3-4