Abstract
Let A be a principally polarized abelian variety of dimension g over a number field k. The Mordell-Weil theorem tells us that the group A(k) of k-rational points of A is finitely generated; in particular, the torsion subgroup of A(k) is finite. For the important special case of an elliptic curve E/Q, Mazur [Mal,2] has proved that the torsion subgroup of E(Q) has order < 16. In general one would expect the torsion subgroup of A(k) to have order < C(g, k), a constant depending only on the number field k and the dimension g but not
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