Supersingular quotients of Fermat curves
For any \(m\geq 4\) let \(F_m\) be the \textit{Fermat curve} \(X^m+Y^m+Z^m=0\) defined over some algebraically closed field \(k\) of characteristic \(p\geq 0\) (assuming \(p\nmid m\)). The goal of the paper is to produce supersingular curves of any genus \(g\geq 2\) in characteristic \(p\neq 2\) by looking at the quotients of Fermat curves by subgroups of their automorphisms groups. Such automorphisms groups are known to be isomorphic to \[ \mu_m^2\rtimes S_3 \;\text{ if }\;\mathrm{char}(k)=0 \;\text{ or }\;\mathrm{char}(k)=p\;\text{ and }\;m\neq p^r+1\,; \] \[ PGU(3,p^{2r}) \;\text{ if }\;\mathrm{char}(k)=p\;\text{ and }\;m = p^r+1\;. \] For the first case in characteristic 0 (but the method works for any characteristic \(\neq 2,3\)) the author uses Riemann existence theorem to study all the quotients of \(F_m\) by subgroups of \(\mu_m^2\) via Galois coverings of \(\mathbb{P}^1\) ramified over three points, then describes the ones admitting an extra action by a subgroup of \(S_3\,\). The case of \(\mathrm{char}(k)=3\) is dealt with by explicit computation. For the second case the author provides a description of the quotients of \(F_m\) by subgroups of certain maximal subgroups of \(PGU(3,p^{2r})\). In the final section all the quotients are then studied in details and careful computations of all the data needed for the Riemann-Hurwitz formula are carried on to finally compute the genus of such quotients. As an example, for \(p=3\) the method produces supersingular curves of any genus \(g\leq 100\) except for \(g=59\) (many other examples are given and can be computed with the formulas provided in the paper).
scientific article; zbMATH DE number 5603985