Jacobi sums, Fermat Jacobians, and ranks of abelian varieties over   towers of function fields
  Our goal in this note is to give a number of examples of abelian varieties over function fields k(t) which have bounded ranks in towers of extensions such as k(t^{1/d}) for varying d. Along the way we prove some new results on Fermat curves which may be of independent interest. 
Douglas Ulmer