On the topology of ℋ(2)

Abstract

The space ℋ(2) consists of pairs (M, ω), where M is a Riemann surface of genus two, and ω is a holomorphic 1-form which has only one zero of order two. There exists a natural action of ℂ* on ℋ(2) by multiplication to the holomorphic 1-form. In this paper, we single out a proper subgroup Γ of Sp(4,ℤ) generated by three elements, and show that the space H(2)/ℂ* can be identified with the quotient Γ\J2, where J2 is the Jacobian locus in the Siegel upper half space H2. A direct consequence of this result is that [Sp(4,ℤ): Γ] = 6. The group Γ can also be interpreted as the image of the fundamental group of H(2)/ℂ* in the symplectic group Sp(4,ℤ). © European Mathematical Society.