Torsion on Abelian Varieties over Large Algebraic Extensions

The Period-index Problem in Wc-groups Ii: Abelian Varieties

We study the relationship between the period and the index of a principal homogeneous space over an abelian variety, obtaining results which generalize work of Cassels and Lichtenbaum on curves of genus one. In addition, we show that the p-torsion in the Shafarevich-Tate group of a fixed abelian variety over a number field k grows arbitarily large when considered over field extensions l/k of bo...

Torsion of abelian varieties over large algebraic fields

Griffiths Groups of Supersingular Abelian Varieties

The Griffiths group Grr(X) of a smooth projective variety X over an algebraically closed field is defined to be the group of homologically trivial algebraic cycles of codimension r on X modulo the subgroup of algebraically trivial algebraic cycles. The main result of this paper is that the Griffiths group Gr(Ak̄) of a supersingular abelian variety Ak̄ over the algebraic closure of a finite field ...

Large Rational Torsion on Abelian Varieties

A method of searching for large rational torsion on Abelian varieties is described. A few explicit applications of this method over Q give rational 11and 13-torsion in dimension 2, and rational 29-torsion in dimension 4.

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 ...