Uniform boundedness of torsion subgroups of linear groups

The torsion conjecture says: for any abelian variety A defined over a number field k, the order of the torsion subgroup of A(k) is bounded by a constant C(k, d) which depends only on the number field k and the dimension d of the abelian variety. The torsion conjecture remains open in general. However, in this paper, a short argument shows that the conjecture is true for more general fields if we consider linear groups instead of abelian varieties. If G is a connected linear algebraic group defined over a field k which is finitely generated over ℚ, Γ is a torsion subgroup of G(k). Then the order of Γ is bounded by a constant C′(k, d) which depends only on k and the dimension d of G. Supported by Tianyuan Mathematics Foundation of NSFC (Grant No. 10626050), and the Knowledge Innovation Program of the Chinese Academy of Sciences