doi:10.1006/jnth.2001.2760 Legendre Elliptic Curves over Finite Fields (original) (raw)
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Legendre Elliptic Curves over Finite Fields
Journal of Number Theory, 2002
We show that every elliptic curve over a finite field of odd characteristic whose number of rational points is divisible by 4 is isogenous to an elliptic curve in Legendre form, with the sole exception of a minimal respectively maximal elliptic curve. We also collect some results concerning the supersingular Legendre parameters. # 2002 Elsevier Science (USA) # 2002 Elsevier Science (USA) All rights reserved.
Designs, Codes and Cryptography, 2015
Supersingular elliptic curves have played an important role in the development of elliptic curve crytography. Scott Vanstone introduced the first author to elliptic curve cryptography, a subject that continues to be a rich source of interesting problems, results and applications. One important fact about supersingular elliptic curves is that the number of rational points is tightly constrained to a small number of possible values. In this paper we present some similar results for curves of higher genus. We also present an application to the problem of determining abelian varieties that occur as jacobians.
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.
Some remarks on the number of points on elliptic curves over finite prime field
Bulletin of the Australian Mathematical Society, 2007
Let p ≥ 5 be a prime and for a, b ε p, let Ea, b denote the elliptic curve over p with equation y2 = x3 + ax + b. As usual define the trace of Frobenius ap, a, b by We use elementary facts about exponential sums and known results about binary quadratic forms over finite fields to evaluate the sums and for primes p in various congruence classes.As an example of our results, we prove the following: Let p ≡ 5 (mod 6) be prime and let b ε *p. Then
On elliptic curves induced by rational Diophantine quadruples
Proc. Japan Acad. Ser. A Math. Sci., 2022
In this paper, we consider elliptic curves induced by rational Dio-phantine quadruples, i.e. sets of four nonzero rationals such that the product of any two of them plus 1 is a perfect square. We show that for each of the groups Z/2Z × Z/kZ for k = 2, 4, 6, 8, there are infinitely many rational Dio-phantine quadruples with the property that the induced elliptic curve has this torsion group. We also construct curves with moderately large rank in each of these four cases.
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] .