Moduli Space of Elliptic Curves with Heisenberg Level Structure

Abstract

A Heisenberg level structure on an elliptic curve is a kind of marking of the linear action by thenoncommutativeHeisenberg group on the space of global sections of a line bundle. The moduli scheme of polarized elliptic curves with Heisenberg level structures is known to be a quasi projective curve over $Z\left[ {{\zeta _d},1/d} \right]$ if the degree of polarization is equal to d ≥3. We prove that it is isomorphic to the (open subscheme of the) moduli schemeX(d)of elliptic curves with level d-structures in the classical sense. Moreover, we also prove that the universal curve over the moduli is the same as that overX(d)ifdis odd, while they are different fordeven. In particular, the universal elliptic curve overSQ⁰(d) has no global sections for d ≥ 4 and even.