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

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.

Theory of optimal harvesting for a size structured model of fish

This paper investigates the maximum principle for a nonlinear size structured model that describes the optimal management of the fish resources taking into account harvesting the fish and putting the fry. First, we show the existence of a unique non-negative solution of the system, and give a comparison principle. Next, we prove the existence of optimal policies by using maximizing sequence and Mazur’s theorem in convex analysis. Then, we obtain necessary optimality conditions by using tangent-normal cones and adjoint system techniques. Finally, some examples and numerical results demonstrate the effectiveness of the theoretical results in our paper.

Rational Poncelet

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.

Counting elliptic curves with prescribed torsion

Mazur’s theorem states that there are exactly fifteen possibilities for the torsion subgroup of an elliptic curve over the rational numbers. We determine how often each of these groups actually occurs. Precisely, if

The differentiability of convex functions on topological linear spaces

In this paper topological linear spaces are categorised according to the differentiability properties of their continuous convex functions. Mazur's Theorem for Banach spaces is generalised: all separable Baire topological linear spaces are weak Asplund. A class of spaces is given for which Gateaux and Fréchet differentiability of a continuous convex function coincide, which with Mazur's theorem, implies that all Montel Fréchet spaces are Asplund spaces. The effect of weakening the topology of a given space is studied in terms of the space's classification. Any topological linear space with its weak topology is an Asplund space; at the opposite end of the topological spectrum, an example is given of the inductive limit of Asplund spaces which is not even a Gateaux differentiability space.