On certain curves of genus three with many automorphisms

Abstract

Let k be an algebraically closed ground field. When C is a complete nonsingular curve of genus g and G is a subgroup of its automorphism group Aut(C), we call the pair (C, G) an AM curve of genus g {AM stands for automorphism ). In Part I, we consider the AM curve (K, Aut(A')), where K is the plane curve defined by XixI+x g+l with g=3. To determine (K, Ant(K)), we use the fact that Aut(C) of a nonsingular quartic plane curve C is canonically identified with a subgroup of PGL(3, k). We shall show in particular that when char(£)=3, (K, Aut(K)) is isomorphic to the AM curve (K4, PSU(3, 32)),where K, is defined by x\+xi+x and PS£7(3,32) is a simple subgroup of PGL(3, k) of order 6048. We note that it is the maximum order among the automorphism groups of (complete nonsingular) curves of genus 3 [8]. In Part II we consider the families of AM curves (C, G) of genus 3, where G is isomorphic to the symmetric group of degree 4, 4. (We note that Aut(K) contains such subgroups.) In §1, we shall determine normal forms of such AM curves. In §2 we shall determine the isomorphism classes in the above normal forms. In §3, using these results, we explain the relations between the subgroups of Teichmiiller modular group Mod(3) which are isomorphic to c4 and their representations on the spaces of holomorphic differentials.In fact,for an AM Riemann surface (W, G) (similarly defined as in the case of AM curves), we obtain naturally a subgroup (denoted by M(W, G)) of the Teichmiiller modular group Mod(3), which is isomorphic to G. Also we obtain a subgroup (denoted by p(W, G)) of GL(3, C) which is the image of the representation of G on the space of holomorphic differentials.We shall prove: