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 Gcovers of the moduli space of configurations of six points on the projective line modulo PGL_2, via the 3 and 2torsion 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 MordellWeil lattice is E_8 with automorphisms by a central extension of G.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 1999
 arXiv:
 arXiv:math/9905195
 Bibcode:
 1999math......5195E
 Keywords:

 Algebraic Geometry;
 Number Theory;
 14H10;
 14D22;
 14H45 (Primary);
 14J25;
 14J26;
 14J27;
 14H40 (Secondary)
 EPrint:
 23 pages