Class numbers and $p$-ranks in ${\mathbb{Z}}_p^d$-towers

To extend Iwasawa’s classical theorem from ${\mathbb{Z}}_p$-towers to ${\mathbb{Z}}_p^d$-towers, Greenberg conjectured that the exponent of $p$ in the $n$-th class number in a ${\mathbb{Z}}_p^d$-tower of a global field $K$ ramified at finitely many primes is given by a polynomial in $p^n$ and $n$ of total degree at most $d$ for sufficiently large $n$. This conjecture remains open for $d\ge 2$. In this paper, we prove that this conjecture is true in the function field case. Further, we propose a series of general conjectures on $p$-adic stability of zeta functions in a $p$-adic Lie tower of function fields.