Title: Genus growth in $\mathbb{Z}_p$-towers of function fields,
Abstract: Let $K$ be a function field over a finite field $k$ of characteristic $p$ and
let $K_{\infty}/K$ be a geometric extension with Galois group $\mathbb{Z}_p$.
Let $K_n$ be the corresponding subextension with Galois group
$\mathbb{Z}/p^n\mathbb{Z}$ and genus $g_n$. In this paper, we give a simple
explicit formula $g_n$ in terms of an explicit Witt vector construction of the
$\mathbb{Z}_p$-tower. This formula leads to a tight lower bound on $g_n$ which
is quadratic in $p^n$. Furthermore, we determine all $\mathbb{Z}_p$-towers for
which the genus sequence is stable, in the sense that there are $a,b,c \in
\mathbb{Q}$ such that $g_n=a p^{2n}+b p^n +c$ for $n$ large enough. Such genus
stable towers are expected to have strong stable arithmetic properties for
their zeta functions. A key technical contribution of this work is a new
simplified formula for the Schmid-Witt symbol coming from local class field
theory.