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Elliptic curves with large torsion and positive rank over number fields of small degree and ECM factorization

By the Mordell-Weil theorem, the group E(K) of K-rational points on an elliptic curve E over a number field K is isomorphic to the product of a finite subgroup consisting of all torsion points and r  0 copies of an in finite cyclic group. By Mazur's theorem, there are exactly 15 possible torsion groups for elliptic curves over Q, but little is known about which values of rank r are possible. Let B(T) = sup{; ; ; rank (E(Q)) : torsion group of E over Q is T}; ; ; . It follows from results of Montgomery, Suyama, Atkin and Morain, motivated by the application of elliptic curves to factorization, that B(T) >= 1 for all admissible torsion groups T. We improved this result by showing that B(T) >= 3 for all T. Recently, we proved similar results for elliptic curves over quadratic, cubic and quartic fields. E.g. we proved that there exist elliptic curves over quadratic elds with positive rank and torsion Z/15Z, Z/2Z * Z/10Z and Z/2Z * Z/12Z. Together with results of Rabarison, this implies that there exist curves with positive rank for all possible torsion groups over quadratic fields (by Kamienny, Kenku and Momose, there are 26 such groups) except maybe for Z=18Z (it seems that for all such curves rank is even). Mazur and Rubin proved recently that for every number field K there exist an elliptic curve over K with rank 0. We show that the statement is not true if one looks at elliptic curves with prescribed torsion over some fixed number field. Namely, we show that all elliptic curves over quartic field Q(i, sqrt(5)) with torsion group Z/15Z have positive rank. In 1987, Lestra proposed the Elliptic curve factorization method (ECM), in which the group F_p*, used in Pollard's p-1 factorization method, is replaced by a group E(F_p), for a suitable chosen elliptic curve E. In ECM, one hopes that the chosen elliptic curve will have smooth order over a prime field. It is now a classical method to use for that purpose elliptic curves E with large rational torsion over Q (and known point of infinite order), as the torsion will inject into E(F_p) for all primes p of good reduction. This in turn makes the order of E(F_p) more likely to be smooth. We will discuss possible applications of elliptic curves with large torsion and positive rank over number fields of small degree (instead over Q) in ECM. Recently, Brier and Clavier used such curves over cyclotomic fields in factorization of Cunningham numbers.

elliptic curves ; number fields ; factorization

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