Some remarks on almost rational torsion points
Journal Article
Overview
abstract
For a commutative algebraic group G over a perfect field k, Ribet defined the set of almost rational torsion points Gtors,kar 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, Ttors,kar⊆T[Ud,g]. We show the corresponding result for abelian varieties with complex multiplication, and under an additional hypothesis, for elliptic curves without complex multiplication. Finally, we show that except for an explicit finite set of semi-abelian varieties G over a finite field k, Gtors,kar is infinite, and use this to show for any abelian variety A over a p-adic field k, there is a finite extension of k over which Ators,kar is infinite.
