Let $K$ be a finite extension of $\mathbb{F}\_q(t)$ and define the curve $C$ by
the equation $y^p=f(x)$ where $p=\mathbf{char} K$ and $f\in K[x]$.
What is the genus of $C$? When does it have infinitely many $K$-points?



Rational points on the curve y^p=f(x) in characteristic p
It "changes genus". It has genus zero over the algebraic closure of $K$ but it behaves as if it had positive genus over $K$ unless there is a change of coordinates over $K$ that changes $f(x)$ into a polynomial defined over $K^p$. When it has positive genus over $K$ in this sense, then the set of $K$-rational points is finite. I proved this in Bull. SMF 119(1991), 121-126.


