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 for $g_n$ 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.