Division by 222 of rational points on elliptic curves (original) (raw)

The divisibility by 2 of rational points on elliptic curves

arXiv: Number Theory, 2017

We give a simple proof of the well-known divisibility by 2 condition for rational points on elliptic curves with rational 2-torsion. As an application of the explicit division by 2n2^n2n formulas obtained in Sec.2, we construct versal families of elliptic curves containing points of orders 4, 5, 6, and 8 from which we obtain an explicit description of elliptic curves over certain finite fields mathbbFq\mathbb{F}_qmathbbFq with a prescribed (small) group E(mathbbFq)E(\mathbb{F}_q)E(mathbbFq). In the last two sections we study 3- and 5-torsion. This paper supercedes arXiv:1605.09279 [math.NT] .

On the torsion of rational elliptic curves over quartic fields

Mathematics of Computation, 2017

Let E be an elliptic curve defined over Q and let G = E(Q)tors be the associated torsion subgroup. We study, for a given G, which possible groups G ⊆ H could appear such that H = E(K)tors, for [K : Q] = 4 and H is one of the possible torsion structures that occur infinitely often as torsion structures of elliptic curves defined over quartic number fields. Let K be a number field, and let E be an elliptic curve over K. The Mordell-Weil theorem states that the set E(K) of K-rational points on E is a finitely generated abelian group. It is well known that E(K) tors , the torsion subgroup of E(K), is isomorphic to Z/nZ × Z/mZ for some positive integers n, m with n|m. In the rest of the paper we shall write C n = Z/nZ for brevity, and we call C n × C m the torsion structure of E over K. The characterization of the possible torsion structures of elliptic curves has been of considerable interest over the last few decades. Since Mazur's proof [36] of Ogg's conjecture, 1 and Merel's proof [37] of the uniform boundedness conjecture, there have been several interesting developments in the case of a number field K of fixed degree d over Q. The case of quadratic fields (d = 2) was completed by Kamienny [29], and Kenku and Momose [31] after a long series of papers. However, there is no complete characterization of the torsion structures that may occur for any fixed degree d > 2 at this time. 2 Nevertheless, there has been significant progress to characterize the cubic case [27, 24, 39, 23, 3, 50] and the quartic case [28, 25, 26, 40]. Let us define some useful notations to describe more precisely what is known for d ≥ 2: • Let S(d) be the set of primes that can appear as the order of a torsion point of an elliptic curve defined over a number field of degree ≤ d. • Let Φ(d) be the set of possible isomorphism torsion structures E(K) tors , where K runs through all number fields K of degree d and E runs through all elliptic curves over K. • Let Φ ∞ (d) be the subset of isomorphic torsion structures in Φ(d) that occur infinitely often. More precisely, a torsion structure G belongs to Φ ∞ (d) if there are infinitely many elliptic curves E, non-isomorphic over Q, such that E(K) tors ≃ G.

Families of elliptic curves with rational 3-torsion

Journal of Mathematical Cryptology, 2012

In this paper we look at three families of elliptic curves with ratio nal 3-torsion over a finite field. These families include Hessian curves, twisted Hessian curves, and a new family we call generalized DIK curves. We find the number of F q-isogeny classes of each family, as well as the number of F q-isomorphism classes of the generalized DIK curves. We also include some formulas for efficient computation on these curves, improving upon known results. In particular, we find better formulas for doubling and addition on the original triplingoriented DIK curves and also for addition and tripling on elliptic curves with j-invariant 0.

Elliptic curves with abelian division fields

Mathematische Zeitschrift, 2016

Let E be an elliptic curve over Q, and let n ≥ 1. The central object of study of this article is the division field Q(E[n]) that results by adjoining to Q the coordinates of all ntorsion points on E(Q). In particular, we classify all curves E/Q such that Q(E[n]) is as small as possible, that is, when Q(E[n]) = Q(ζn), and we prove that this is only possible for n = 2, 3, 4, or 5. More generally, we classify all curves such that Q(E[n]) is contained in a cyclotomic extension of Q or, equivalently (by the Kronecker-Weber theorem), when Q(E[n])/Q is an abelian extension. In particular, we prove that this only happens for n = 2, 3, 4, 5, 6, or 8, and we classify the possible Galois groups that occur for each value of n.