The number of genus 2 covers of an elliptic curve

The main aim of this paper is to determine the number c(N,D) of genus 2 covers of an elliptic curve E of fixed degree N >= 1 and fixed discriminant divisor D is an element of Div (E). In the case that D is reduced, this formula is due to Dijkgraaf. The basic technique here for determining c(N,D) is to exploit the geometry of a certain compactification C=C(E,N) of the universal genus 2 curve over the Hurwitz space H(E,N) which classifies (normalized) genus 2 covers of degree N of E. Thus, a secondary aim of this paper is to study the geometry of C. For example, the structure of its degenerate fibres is determined, and this yields formulae for the numerical invariants of C which are also of independent interest.