Mordell-Weil Groups of Large Rank in Towers

Abstract

Let k be the algebraic closure of the field with q elements. We build upon recent work of Ulmer and Berger to give examples of elliptic curves and higher dimensional abelian varieties over the field K=k(t) with the property that their ranks become arbitrarily large when dth roots of the variable t are adjoined to K for d varying across the integers relatively prime to q. We also give a first example of an elliptic curve whose rank under such extensions grows linearly in d, for those d prime to q.