Let $q=p^n$. Let $C$ be an Artin schierer curve defined by $y^py=f(x)$ where $f(x) \in \mathbb{F}_{q}[x]$. Let $C_g$ be $y^py=f(x)$ where $x \in g(\mathbb{F}_{p^{n}})$ for some $g \in \mathbb{F}_q[x]$. Is there a relation between number of solutions of $C$ and $C_g$? Or Zeta function of $C_g$ and $C$ ?
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