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Let $q=p^n$, for $p$ a prime. Let $C$ be an Artin–Schreier curve defined by $y^p-y=f(x)$ where $f(x) \in \mathbb{F}_{q}[x]$.

Let $C_g$ be $y^p-y=f(x)$ 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 $C_g$? Is there a Zeta function for $C_g$ and $C$ ?

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    $\begingroup$ I don't really understand your notation. Is $C_g$ the curve $y^p-y=f(g(x))$? If not, then what? $\endgroup$
    – Donu Arapura
    Commented Feb 18, 2012 at 16:10
  • $\begingroup$ Yes, but I am looking for solutions over $g(\mathbb{F}_q)$ not over $\mathbb{F}_q$. Please let me know if its still unclear. Thanks! $\endgroup$
    – Heinrich
    Commented Feb 18, 2012 at 16:34
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    $\begingroup$ Expanding on Donu's comment, $C_g: y^p-y=f(g(x))$ maps to $C: y^p-y=f(x)$ by $(x,y) \mapsto (g(x),y)$. From general theory, it follows that the numerator of the zeta function of $C$ divides the numerator of the zeta function of $C_g$. One can also describe the number of points of $g(\mathbb{F}_q)$ in terms of the Galois group of $g(x)-t$, but the answer will depend a lot on $g$ and to get results about $C$ and $C_g$ you'll need to know whether $f$ and $g$ are related or not. Do you have a specific case that you are interested in? $\endgroup$
    – Felipe Voloch
    Commented Feb 18, 2012 at 21:44
  • $\begingroup$ Thanks Felipe!It really helped. Can you please suggest me a reference to read on this? I was interested in the case when $g(x)=x+\frac{1}{x}$ and $f$ is any polynomial. $\endgroup$
    – Heinrich
    Commented Feb 19, 2012 at 6:33
  • $\begingroup$ Birch and Swinnerton-Dyer, Note on a problem of Chowla, Acta Arith, 5 (1959) 417-423, for value sets of general $g$. But for $x+1/x$ you won't need this. $\endgroup$
    – Felipe Voloch
    Commented Feb 19, 2012 at 16:21

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