Abelian Varieties over Large Algebraic Fields with Infinite Torsion

Torsion of abelian varieties over large algebraic fields

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...

Addendum to: "Infinite-dimensional versions of the primary, cyclic and Jordan decompositions", by M. Radjabalipour

In his paper mentioned in the title, which appears in the same issue of this journal, Mehdi Radjabalipour derives the cyclic decomposition of an algebraic linear transformation. A more general structure theory for linear transformations appears in Irving Kaplansky's lovely 1954 book on infinite abelian groups. We present a translation of Kaplansky's results for abelian groups into the terminolo...

Torsion of Abelian Varieties

We prove: Let A be an abelian variety over a number field K. Then K has a finite Galois extension L such that for almost all σ ∈ Gal(L) there are infinitely many prime numbers l with Al(K̃(σ)) 6= 0. Here K̃ denotes the algebraic closure of K and K̃(σ) the fixed field in K̃ of σ. The expression “almost all σ” means “all but a set of σ of Haar measure 0”. MR Classification: 12E30 Directory: \Jarden\D...

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.