Genus Stability of ${\mathbb{Z}}_p^{\times }$-Towers

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