Modular invariant Virasoro modules and elliptic curves

Abstract

Given a rational number ζ, when does the tensor product of two modular invariant Virasoro modules have central charge ζ? I show that the resulting equations define elliptic curves over the rational numbers, and determine the torsion subgroups of these curves for the case in which both tensor factors have the same central charge. This gives an alternative parametrization of the curves with torsion subgroup ℤ/8ℤ x ℤ/2ℤ. Consideration of a larger class of curves (including some which do not correspond to tensor products of modules) gives a parametrization of all curves with a point of order 8. I conclude with some examples of curves which have 16 torsion points and positive rank.