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

TL;DR

This paper proves Greenberg's conjecture for class numbers in ${ m Z}_p^d$-towers over function fields and discusses broader conjectures on p-adic stability of zeta functions in p-adic Lie towers.

Contribution

It establishes the conjecture in the function field case and introduces new conjectures on p-adic stability of zeta functions in p-adic Lie towers.

Findings

Greenberg's conjecture holds in the function field case for ${ m Z}_p^d$-towers.

Proposes general conjectures on p-adic stability of zeta functions.

Provides evidence supporting polynomial behavior of class number exponents.

Abstract

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 $d\geq 2$. 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.