Hurwitz spaces of quadruple coverings of elliptic curves and the modulispace of abelian threefolds 𝒜 3 (1, 1, 4)

We prove that the moduli space 3 (1, 1, 4) of polarized abelian threefolds with polarization of type (1, 1, 4) is unirational. By a result of Birkenhake and Lange this implies the unirationality of the isomorphic moduli space 3 (1, 4, 4). The result is based on the study the Hurwitz space ℋ 4 ,n ( Y ) of quadruple coverings of an elliptic curve Y simply branched in n ≥ 2 points. We prove the unirationality of its codimension one subvariety ℋ 0 4 ,A ( Y ) which parametrizes quadruple coverings π : X → Y with Tschirnhausen modules isomorphic to A –1 , where A ∈ Pic n /2 Y , and for which π * : J ( Y ) → J ( X ) is injective. This is an analog of the result of Arbarello and Cornalba that the Hurwitz space ℋ 4 ,n (ℙ 1 ) is unirational. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)