On certain curves of genus three with many automorphisms (Q790197)

This paper consists of two parts. In the first part the author studies the symmetries of the Klein quartic given as a plane curve in \({\mathbb{P}}^ 2_ k\) by the equation \(x_ 0x_ 1^ 3+x_ 1x_ 2^ 3+x_ 2x_ 0^ 3=0\). At least if \(char(k)=0\) it is well known that this curve admits an automorphism group of order 168 which is isomorphic to PSL(2,7). The author treats in some detail the cases where \(char(k)=2\) or 3. In the second case the curve has an automorphism group of order 6048. In the second part the author considers curves of genus 3 whose group of automorphisms is isomorphic to the symmetric group \(S_ 4\). He describes normal forms for such curves, both in the case where C is hyperelliptic and in the case where the canonical image of C is a plane quartic. He finally relates this to subgroups of the Teichmüller modular group Mod(3) which are isomorphic to \(S_ 4\) and their canonical representations on the space of holomorphic differentials.