Algebraic approximations of fibrations in abelian varieties over a curve

Torsion of abelian varieties over large algebraic fields

Abelian Varieties over Large Algebraic Fields with Infinite Torsion

Let A be a non-zero abelian variety defined over a number field K and let K be a fixed algebraic closure of K. For each element σ of the absolute Galois group Gal(K/K), let K(σ) be the fixed field in K of σ. We show that the torsion subgroup of A(K(σ)) is infinite for all σ ∈ Gal(K/K) outside of some set of Haar measure zero. This proves the number field case of a conjecture of W.-D. Geyer and ...

Torsion on Abelian Varieties over Large Algebraic Extensions

Let K be a finitely generated extension of Q and A a non-zero abelian variety over K. Let K̃ be the algebraic closure of K and Gal(K) = Gal(K̃/K) the absolute Galois group of K equipped with its Haar measure. For each σ ∈ Gal(K) let K̃(σ) be the fixed field of σ in K̃. We prove that for almost all σ ∈ Gal(K) there exist infinitely many prime numbers l such that A has a non-zero K̃(σ)-rational point ...

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...

Abelian varieties over finite fields

A. Weil proved that the geometric Frobenius π = Fa of an abelian variety over a finite field with q = pa elements has absolute value √ q for every embedding. T. Honda and J. Tate showed that A 7→ πA gives a bijection between the set of isogeny classes of simple abelian varieties over Fq and the set of conjugacy classes of q-Weil numbers. Higher-dimensional varieties over finite fields, Summer s...