Genus Stability of $\mathbb Z_p^\times$-Towers

TL;DR

This paper investigates the genus growth in towers of algebraic curves over finite fields with Galois group isomorphic to _p^ imes, identifying all towers with quadratic genus growth in terms of p^n for large n.

Contribution

It characterizes all _p^ imes-towers of curves with quadratic genus growth, providing a complete classification for large n.

Findings

Identifies all _p^ imes-towers with quadratic genus growth.

Determines the conditions under which genus growth follows a quadratic pattern.

Provides explicit descriptions of such towers.

Abstract

Let $p$ be a prime. Consider a tower of smooth projective geometrically irreducible curves over $\mathbb F_p$, $\mathscr C:\cdots\rightarrow C_n\rightarrow\cdots\rightarrow C_1\rightarrow C_0=\mathbb P^1$ whose Galois group is isomorphic to $\mathbb Z_p^\times$. In this paper, we study genus growth of the tower $\mathscr C$ and determine all the $\mathbb Z_p^\times$-towers with genus be a quadratic equation of $p^{n}$ when $n$ is sufficiently large.