Rational points of Abelian varieties in Γ-extension

Density Measure of Rational Points on Abelian Varieties

Let A be a simple Abelian variety of dimension g over , and let ` be the rank of the Mordell-Weil group A( ). Assume ` ≥ 1. A conjecture of Mazur asserts that the closure of A( ) into A( ) for the real topology contains the neutral component A( ) of the origin. This is known only under the extra hypothesis ` ≥ g − g + 1. We investigate here a quantitative refinement of this question: for each g...

Groups of Rational Points on Abelian Varieties over Finite Fields

Fix an isogeny class of abelian varieties with commutative endomorphism algebra over a finite field. This isogeny class is determined by a Weil polynomial fA without multiple roots. We give a classification of groups of rational points on varieties from this class in terms of Newton polygons of fA(1− t).

Rational points on varieties

Abelian Points on Algebraic Varieties

We attempt to determine which classes of algebraic varieties over Q must have points in some abelian extension of Q. We give: (i) for every odd d > 1, an explicit family of degree d, dimension d − 2 diagonal hypersurfaces without Qab-points, (ii) for every number field K, a genus one curve C/Q with no K ab-points, and (iii) for every g ≥ 4 an algebraic curve C/Q of genus g with no Qab-points. I...

Sequences of Rational Torsions on Abelian Varieties

We address the question of how fast the available rational torsion on abelian varieties over Q increases with dimension. The emphasis will be on the derivation of sequences of torsion divisors on hyperelliptic curves. Work of Hellegouarch and Lozach (and Klein) may be made explicit to provide sequences of curves with rational torsion divisors of orders increasing linearly with respect to genus....