How does the rank of an elliptic curve grow in towers of number fields?
On an elliptic curve y2=x3+ax+b, the points with coordinates (x,y) in a given number field form a finitely generated abelian group. One natural question is how the rank of this group changes when changing the number field.
One may guess that the rank should keep growing in this tower ('more numbers mean more solutions'). However, this guess turns out to be incorrect -- the rank is always bounded, as envisioned by the theories of Iwasawa and Mazur in the 1970's.