On a family of quotients of Fermat curves (Q1904463)

Let \(F_N\) be the Fermat curve defined by: \(u^N + v^N=1\) and let \(r,s \in \mathbb{N}\) be such that \(r+s \leq N+1\) and g.c.d.\((r,s,N) = 1\); the curve \(F(r,s)\) is the quotient of \(F_N\) defined by the equation: \(y^N = x^r(1-x)^s\), where the projection is defined by \((x,y) = (u^N, u^rv^s)\). The genus of \(F(r,s)\), \(g(r,s)\), is such that: \(N \geq 2g (r,s)+1\); in this paper it is shown that this inequality characterizes the curves possessing an automorphism of order \(N\) (for \(F(r,s)\) the automorphism \((x,y) \to (x,\xi_Ny)\), where \(\xi_N\) is a \(N\)-th root of the unity, has order \(N)\). In fact it is shown that a curve \(X\), over an algebraically closed field \(k\) with \(\text{char} k=0\), possessing an automorphism of order \(N\geq 2g + 1\) is either a hyperelliptic curve \(H_\lambda\) defined by \(y^2 = (x^{g+1} - 1) (x^{g+1} - \lambda)\), \(\lambda \in k \backslash \{0,1\}\) (in this case \(N-2 g+2)\), or it is of type \(F(r,s)\). -- Then the curves for which \(N=2 g+1\) are studied; among them hyperelliptic curves and curves for which the automorphism group is cyclic of maximal order are determined (i.e. their isomorphism class are determined).