Some Rank Records for Elliptic Curves with Prescribed Torsion Over Quadratic Fields

Najman, Filip

doi:10.2478/auom-2014-0017

Abstract

In this paper we construct elliptic curves over quadratic fields with prescribed torsion group and record rank. We do this using theoretical arguments and keeping lengthy computations to a minimum.

References

[1] W. Bosma, J. J. Cannon, C. Fieker, A. Steel (eds.), Handbook of Magma functions, Edition 2.18 (2012).
Search in Google Scholar
[2] J. Aguirre, A. Dujella, M. Jukic Bokun, J. C. Peral, High rank elliptic curves with prescribed torsion group over quadratic fields, Period. Math. Hungar., to appear.
Search in Google Scholar
[3] J. Bosnian, P. Bruin, A. Dujella, F. Najman, Ranks of elliptic curves with prescribed torsion over number fields, Int. Math. Res. Not. IMRN, to appear.
Search in Google Scholar
[4] A. Dujella, High rank elliptic curves with prescribed torsion, http://web. math.hr/~duje/tors/tors.html
Search in Google Scholar
[5] A. Dujella, M. Jukić Bokun, On the rank of elliptic curves overQ(i) with torsion group Z4 ⊕ Z4, Proc. Japan Acad. Ser. A Math. Sci. 86 (2010), 93-96.10.3792/pjaa.86.93
Search in Google Scholar
[6] A. Dujella, F. Najman, Elliptic curves with large torsion and positive rank over number fields of small degree and ECM factorization, Period. Math. Hungar. 65 (2012), 193-203.10.1007/s10998-012-8257-7
Search in Google Scholar
[7] N. Elkies, j = 0, rank 15; also 3-rank 6 and 7 in real and imaginary quadratic fields, Number Theory Listserv posting, December 30, 2009, http://listserv.nodak.edu/cgi-bin/wa.exe?A2= ind0912&L=NMBRTHRY&F=&S=&P=14012
Search in Google Scholar
[8] N. D. Elkies, N. F. Rogers, Elliptic curves x³ + y³ = k of high rank, in Algorithmic number theory (ANTS-VI), ed. D. Buell, Lecture Notes in Comput. Sci. 3076, Springer, Berlin, 2004, 184-193.
Search in Google Scholar
[9] M. Jukic Bokun, On the rank of elliptic curves over Q(√−3) with torsion group Z₃⊕Z₃and Z₃⊕Z₆, Proc. Japan Acad. Ser. A Math. Sci. 87 (2011), 61-64.10.3792/pjaa.87.61
Search in Google Scholar
[10] M. Laska and M. Lorenz, Rational points on elliptic curves over Qin elementary abelian 2-extensions of Q, J. Reine Angew. Math. 355 (1985), 163-172.
Search in Google Scholar
[11] F. P. Rabarison, Structure de torsion des courbes elliptiques sur les corps quadratiques, Acta Arith. 144 (2010), 17-52.10.4064/aa144-1-3
Search in Google Scholar

Some Rank Records for Elliptic Curves with Prescribed Torsion Over Quadratic Fields

Abstract

Paradigm

My account