Some remark on rational points

Some Remarks on Almost Rational Torsion Points

For a commutative algebraic group G over a perfect field k, Ribet defined the set of almost rational torsion points G tors,k of G over k. For positive integers d, g, we show there is an integer Ud,g such that for all tori T of dimension at most d over number fields of degree at most g, T ar tors,k ⊆ T [Ud,g]. We show the corresponding result for abelian varieties with complex multiplication, an...

Rational Points on Some Fano Quadratic Bundles

We study the number of rational points of bounded height on a certain threefold. The accumulating subvarieties are Zariski-dense in this example. The computations support an extension of a conjecture of Manin to this situation.

Some remarks on almost rational torsion points par

Let G be a commutative algebraic group defined over a perfect field k. Let k be an algebraic closure of k and Γk be the Galois group of k over k. Following Ribet ([1], [19], see also [7]), we say a point P ∈ G(k) is almost rational over k if whenever σ, τ ∈ Γk are such that σ(P )+ τ(P ) = 2P , then σ(P ) = τ(P ) = P . We denote the almost rational points of G over k by Gar k . Let Gtors denote ...

Covering Techniques and Rational Points on Some Genus 5 Curves

We describe a method that allows, under some hypotheses, to compute all the rational points of some genus 5 curve defined over a number field. This method is used to solve some arithmetic problems that remained open.

Rational Points on Quartics