Poncelet problem for rational conics

Малышев, В. А.

doi:10.1090/s1061-0022-08-01012-1

Cited by 5 publications

(4 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…So this composition has order d precisely when the point σ (τ (P)) has order d in the group X(K ) with unit element P. This observation together with Theorem 1.1 explains and improves a result by V.A. Malyshev [9,Introduction]; we will see how with a small additional argument it also implies a result by B. Mirman [11,Cor. 3.5].…”

Section: Examples Starting From Involutionssupporting

confidence: 69%

Mazur’s rational torsion result for pointless genus one curves: examples

2021

View full text Add to dashboard Cite

This note reformulates Mazur’s result on the possible orders of rational torsion points on elliptic curves over $$\mathbb {Q}$$ Q in a way that makes sense for arbitrary genus one curves, regardless whether or not the curve contains a rational point. The main result is that explicit examples are provided of ‘pointless’ genus one curves over $$\mathbb {Q}$$ Q corresponding to the torsion orders 7, 8, 9, 10, 12 (and hence, all possibilities) occurring in Mazur’s theorem. In fact three distinct methods are proposed for constructing such examples, each involving different in our opinion quite nice ideas from the arithmetic of elliptic curves or from algebraic geometry.

show abstract

Section: Examples Starting From Involutionssupporting

confidence: 69%

Mazur’s rational torsion result for pointless genus one curves: examples

2021

View full text Add to dashboard Cite

show abstract

“…The proof of this result presented below is constructive, so in particular it allows one to present explicit K-rational examples of Poncelet's closure theorem whenever an elliptic curve with the required properties exists over K. In case K = Q this happens for n ∈ We now briefly discuss some related work on rationality issues for Poncelet's closure theorem. The paper [15] by V. A. Malyshev discusses the problem of finding the integers n such that two conic sections C, D defined over Q exist, yielding purely periodic Poncelet sequences of exact period n. However, he does not demand that the points P j (and therefore the lines j = P j P j+1 as well) are defined over Q. e.g., the third example he presents in his paper uses C, D given by x 2 +y 2 = 4 and x 2 +y 2 = 3, respectively.…”

Section: Introductionmentioning

confidence: 99%

“…. The example Malyshev presents on the last page of [15] indeed is a purely periodic Q-rational Poncelet sequence of exact period 5.…”

Section: Introductionmentioning

confidence: 99%

“…As a consequence, it is not immediate how to use arithmetic properties of elliptic curves over Q in order to draw conclusions about the arithmetic of the curve X. The following result presents an example of this, in fact used in the papers [15,17] although not explicitly stated there. Since the cited references do not contain a detailed proof of it, we offer one here.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Rational Poncelet

Los

Mepschen

Top

2018

Int. J. Number Theory

View full text Add to dashboard Cite

We construct rational Poncelet configurations, which means finite sets of pairwise distinct [Formula: see text]-rational points [Formula: see text] in the plane such that all [Formula: see text] are on a fixed conic section defined over [Formula: see text], and moreover the lines [Formula: see text] are all tangent to some other fixed conic section defined over [Formula: see text]. This is done for [Formula: see text] in which case only [Formula: see text] and [Formula: see text] are possible, and for certain real quadratic number fields [Formula: see text]; here moreover [Formula: see text] and [Formula: see text] and [Formula: see text] occur, but no further new values of [Formula: see text]. In fact, for every pair [Formula: see text] presented here, we show that infinitely many such tuples [Formula: see text] exist. The construction uses elliptic curves [Formula: see text] over [Formula: see text] such that the group [Formula: see text] is infinite and moreover contains a point of exact order [Formula: see text]. As an aside, a formulation of Mazur’s theorem/Ogg’s conjecture in terms of arbitrary genus one curves over the rational numbers (so not necessarily containing any rational point) is presented, since this occurs naturally in the context of Poncelet configurations.

show abstract

Short cycles of Poncelet’s conics

Mirman

2010

Linear Algebra and its Applications

View full text Add to dashboard Cite

Poncelet problem for rational conics

Abstract: Abstract. The Poncelet problem for rational conics may have a solution only for

Cited by 5 publications

References 2 publications

Mazur’s rational torsion result for pointless genus one curves: examples

Mazur’s rational torsion result for pointless genus one curves: examples

Rational Poncelet

Short cycles of Poncelet’s conics

Contact Info

Product

Resources

About