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In this paper we prove the Geyer-Jarden conjecture on the torsion part of theMordell-Weil group for a large class of abelian varieties defined over finitelygenerated fields of arbitrary characteristic. The class consists of all abelianvarieties with big monodromy, i.e., such that the image of Galoisrepresentation on l-torsion points, for almost all primes l, contains the fullsymplectic group.

Abelian varieties over finitely generated fields and the conjecture of Geyer and Jarden on torsion