The identification of three moduli spaces

Abstract

It is one of the wonderful ``coincidences'' of the theory of finite groups that the simple group G of order 25920 arises as both a symplectic group in characteristic 3 and a unitary group in characteristic 2. These two realizations of G yield two G-covers of the moduli space of configurations of six points on the projective line modulo PGL_2, via the 3- and 2-torsion of the Jacobians of the double and triple cyclic covers of P^1 branched at those six points. Remarkably these two covers are isomorphic. This was proved over C by transcendental methods by Hunt and Weintraub. We give an algebraic proof valid over any field not of characteristic 2 or 3 that contains the cube roots of unity. We then explore the connection between this $G$-cover and the elliptic surface $y^2 = x^3 + sextic(t), whose Mordell-Weil lattice is E_8 with automorphisms by a central extension of G.