# Genus growth in $\mathbb{Z}_p$-towers of function fields

## TL;DR

This paper provides an explicit formula for the genus growth in $Z_p$-towers of function fields, characterizes stable towers with quadratic genus growth, and introduces a simplified Schmid-Witt symbol formula.

## Contribution

It introduces a simple explicit genus formula for $Z_p$-towers and characterizes all stable genus towers, advancing understanding of their arithmetic properties.

## Key findings

- Derived a quadratic lower bound for genus growth in $Z_p$-towers.
- Identified conditions for genus stability with quadratic polynomial growth.
- Developed a new simplified formula for the Schmid-Witt symbol.

## 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.