Rational Poncelet

Abstract

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.

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

“…in which the successive extensions have degree 2. It is easy to obtain a generator of the leftmost field, e.g., using [8,Lemma 2.1]. Using a basis (such as {1, x, √ 13, x √ 13}) of the rightmost field as vector space over the leftmost one and the description of the linear map ι in terms of this basis, a straightforward calculation results in an equation…”

“…A slightly weaker statement (formulated in the language of function fields) is given in [8,Corollary 1.3]. In the special case of Theorem 1.1 that the curve X contains a rational point P (so, (X, P) is an elliptic curve), a proof is given in the paragraph preceding the statement of the theorem.…”