Naslov
Construction of high-rank elliptic curves with non-trivial torsion group

Autori
Dujella, Andrej

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, sažetak, znanstveni

Izvornik
Second Central European Conference on Cryptography HAJDUCRYPT'02 / Petho, Attila - Deberecen : University of Debrecen, 2002, 6-6

Skup
Second Central European Conference on Cryptography HAJDUCRYPT'02

Mjesto i datum
Debrecen, Mađarska, 04.07.2002. - 06.07.2002

Vrsta sudjelovanja
Predavanje

Vrsta recenzije
Međunarodna recenzija

Ključne riječi
elliptic curve; rank; torsion group

Sažetak
The group of an elliptic curve over the rationals is the product of the torsion group and r copies of infinite cyclic group. By the famous theorem of Mazur, there are exactly 15 possible torsion groups. On the other hand, very little is known about which values of rank r are possible. The conjecture is that rank can be arbitrary large, but at present only an example of elliptic curve over Q with rank >= 24 is known. There is even a stronger conjecture that for any of 15 possible torsion groups T we have B(T) = infinity, where B(T)= sup {rank(E(Q)) : torsion group of E over Q is T}. In construction of examples with high rank and prescribed torsion group, we started with families with relatively high generic rank. The next step is to choose, in given family, the best candidates for highest rank. This can be done using Meste-Nagao method, i.e. computing sums for which it is experimentally known that we may expect that they are large for high-rank curves. In the computation of the ranks, we used program MWRANK if the torsion group has even order. For curves with torsion groups of odd order, we used program RATPOINTS in search for independent points of infinite order. This gives us an lower bound for the rank. The upper bound is obtained using the Mazur's bound for the rank of elliptic curves with nontrivial torsion (implemented in APECS). By our methods, we were able to find the record curves in 13 of 15 categories (some of these results are joint work with L. Kulesz and O. Lecacheux). It follows from results of Montgomery and Atkin & Morain (motivated by finding curves suitable for the elliptic curve method of factorization) that B(T) >= 1 for all admissible torsion groups T. We improved this result by showing that B(T) >= 3 for all T. In particular, we found an elliptic curve with rank = 15, what is at present the highest known rank for curves with non-trivial torsion, and also the highest exactly (not just a lower bound) computed rank.

Izvorni jezik
Engleski

Znanstvena područja
Matematika, Računarstvo

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