Torsion points on elliptic curves (original) (raw)

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.

On the group orders of elliptic curves over finite fields (Compositio Mathematica 85 (1993) 229-247)

Given a prime power q, for every pair of positive integers m and n with m|gcd(n, q − 1) we construct a modular curve over Fq that parametrizes elliptic curves over Fq along with Fq-defined points P and Q of order m and n, respectively, with P and n mQ having a given Weil pairing. Using these curves, we estimate the number of elliptic curves over Fq that have a given integer N dividing the number of their Fq-defined points.

Ranks of elliptic curves with prescribed torsion over number fields

Int. Math. Res. Not. IMRN, 2014

We study the structure of the Mordell-Weil group of elliptic curves over number fields of degree 2, 3, and 4. We show that if T is a group, then either the class of all elliptic curves over quadratic fields with torsion subgroup T is empty, or it contains curves of rank 0 as well as curves of positive rank. We prove a similar but slightly weaker result for cubic and quartic fields. On the other hand, we find a group T and a quartic field K such that among the elliptic curves over K with torsion subgroup T , there are curves of positive rank, but none of rank 0. We find examples of elliptic curves with positive rank and given torsion in many previously unknown cases. We also prove that all elliptic curves over quadratic fields with a point of order 13 or 18 and all elliptic curves over quartic fields with a point of order 22 are isogenous to one of their Galois conjugates and, by a phenomenon that we call false complex multiplication, have even rank. Finally, we discuss connections with elliptic curves over finite fields and applications to integer factorization.

Multiples of integral points on elliptic curves

Journal of Number Theory, 2009

If E is a minimal elliptic curve defined over Z, we obtain a bound C, depending only on the global Tamagawa number of E, such that for any point P ∈ E(Q), nP is integral for at most one value of n > C. As a corollary, we show that if E/Q is a fixed elliptic curve, then for all twists E ′ of E of sufficient height, and all torsion-free, rank-one subgroups Γ ⊆ E ′ (Q), Γ contains at most 6 integral points. Explicit computations for congruent number curves are included.

Elliptic curves with torsion groups ℤ/8ℤ and ℤ/2ℤ×ℤ/6ℤ

2021

In this paper, we present details of seven elliptic curves over ℚ(u) with rank 2 and torsion group ℤ/ 8ℤ and five curves over ℚ(u) with rank 2 and torsion group ℤ/ 2ℤ×ℤ/ 6ℤ. We also exhibit some particular examples of curves with high rank over ℚ by specialization of the parameter. We present several sets of infinitely many elliptic curves in both torsion groups and rank at least 3 parametrized by elliptic curves having positive rank. In some of these sets we have performed calculations about the distribution of the root number. This has relation with recent heuristics concerning the rank bound for elliptic curves by Park, Poonen, Voight and Wood.

The uniform primality conjecture for elliptic curves

Acta Arithmetica, 2008

An elliptic divisibility sequence, generated by a point in the image of a rational isogeny, is shown to possess a uniformly bounded number of prime terms. This result applies over the rational numbers, assuming Lang's conjecture, and over the rational function field, unconditionally. In the latter case, a uniform bound is obtained on the index of a prime term. Sharpened versions of these techniques are shown to lead to explicit results where all the irreducible terms can be computed.

On the torsion group of elliptic curves induced by Diophantine triples over quadratic fields

Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM , 2017

The possible torsion groups of elliptic curves induced by Diophan-tine triples over quadratic fields, which do not appear over Q, are Z/2Z × Z/10Z, Z/2Z × Z/12Z and Z/4Z × Z/4Z. In this paper, we show that all these torsion groups indeed appear over some quadratic field. Moreover, we prove that there are infinitely many Diophantine triples over quadratic fields which induce elliptic curves with these tor-sion groups.