Abstract
Let A be an abelian variety of dimension g ⩾ 1 defined over a number field K. We study the size of the torsion group A ( F ) tors where F / K is a finite extension and more precisely we study the best possible exponent γ in the inequality Card ( A ( F ) tors ) ≪ [ F : K ] γ when F is any finite extension of K. In the CM case we give an exact formula for the exponent γ in terms of the characters of the Mumford–Tate group—a torus in this case—and discuss briefly the general case. Finally we give an application of the main result in direction of a generalisation of the Manin–Mumford conjecture.
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