I will report on my recent joint work with S. Arias-de-Reyna and S. Petersen. We have proven conjecture of Geyer and Jarden on torsion part of the Mordell-Weil group for abelian varieties with big monodromy. By definition, an abelian variety over a fin. gen. field has big monodromy, if the image of Galois representation attached to its l-torsion points contains the full symplectic group, for almost all primes l. In addition, we have also found a new, large family of abelian varieties with big monodromy defined over fin. gen. fields of arbitrary characteristic. In the first part of the talk I'm going to give an up-to-date survey on monodromy computations for abelian varieties (in the above sense) and discuss some interesting applications both to geometry and arithmetic.

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