I'm trying to find information on superelliptic curves and how to solve them over the integers. The equation is $$y^k = f(x)$$ where $k=3$ and $f$ has degree $d=3$. Does anyone know any references/books/links that may help me? I'm finding a few PDF's but they all assume $gcd(k,d)=1$.
2026-03-30 12:18:54.1774873134
Superelliptic curves
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What do you mean "solve them over integers"?
The assumption gcd(k, d)=1 is simply to assure that the degree d cyclic covering is tame. Usually, people will call a curve "superelliptic" if it is of the form $$y^k = f(x)$$ for any degree $d > 3$ polynomial $f(x)$.