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 ar,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 multi- plication, 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, G ar,k 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 A ar,k is infinite.