For the simplest example with infinitely many number fields, fix a prime $p$. Adjoining to $\mathbf{Q}$ the $p$th, $p^2$th, $p^3$th,... roots of unity produces a **tower** of number fields \[\mathbf{Q} \subset \mathbf{Q}(\zeta_p)\subset \mathbf{Q}(\zeta_{p^2})\subset .... \] 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.

The above tower started with $\mathbf{Q}$, but there are analogous towers that start with an imaginary quadratic field instead. Given the above boundedness result, one would now guess that the rank is bounded in these towers, too. Surprisingly, this is not the case -- there are scenarios both for bounded and unbounded rank. So how does the rank grow in those towers in general? We initiate an answer to this question in this talk. This is joint work with Antonio Lei.