The identification of three moduli spaces

TL;DR

This paper proves an algebraic isomorphism between two G-covers of the moduli space of six points on the projective line, arising from different group realizations, and explores their connection to an elliptic surface with an E_8 lattice.

Contribution

It provides an algebraic proof of the isomorphism between two moduli space covers over fields with certain roots of unity, extending previous transcendental results.

Findings

Two G-covers of the moduli space are isomorphic.

The algebraic proof applies over any suitable field, not just over complex numbers.

Connection established between the G-cover and an elliptic surface with E_8 lattice.

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.