Abstract
Let 
be a smooth projective curve defined over a number field k, A/k() an abelian variety and (τ, B) the k()/k-trace of A. We estimate how the rank of A(k())/τB(k) varies when we take a finite geometrically abelian cover 
defined over k.
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This work was partially supported by CNPq research grant 304424/2003-0, Pronex 41.96.0830.00 and CNPq Edital Universal 470099/2003-8. I would like to thank Douglas Ulmer for comments on how to treat the case of arbitrary ramification, but the conductor prime to the ramification locus, in the case of elliptic fibrations. I would also like to thank Marc Hindry for comments on the inequality comparing the conductors of A and A'. Finally, I also thank the referee for his comments and criticisms.
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Pacheco, A. On the rank of abelian varieties over function fields. manuscripta math. 118, 361–381 (2005). https://doi.org/10.1007/s00229-005-0597-7
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DOI: https://doi.org/10.1007/s00229-005-0597-7