The sextic plane curves that are invariant under the standard action of the icosahedral group on the projective plane make up a pencil of genus ten curves (spanned by a sum of six lines and a three times a conic). This pencil was first considered in a note by R. M. Winger in 1925 and is nowadays named after him. The second author recently gave this a modern treatment and proved among other things that it contains essentially every smooth genus ten curve with icosahedral symmetry. We here show that the Jacobian of such a curve contains the tensor product of an elliptic curve with a certain integral representation of the icosahedral group. We find that the elliptic curve comes with a distinguished point of order 3, prove that the monodromy on this part of the homology is the full congruence subgroup Γ 1 (3) ⊂ SL 2 (Z) and subsequently identify the base of the pencil with the associated modular curve.We also observe that the Winger pencil 'accounts' for the deformation of the Jacobian of Bring's curve as a principal abelian fourfold with an action of the icosahedral group.

No abstract

The sextic plane curves that are invariant under the standard action of the icosahedral group on the projective plane make up a pencil of genus ten curves (spanned by a sum of six lines and a three times a conic). This pencil was first considered in a note by R. M. Winger in 1925 and is nowadays named after him. We gave this a modern treatment and proved among other things that it contains essentially every smooth genus ten curve with icosahedral symmetry. We here consider the monodromy group and the period map naturally defined by the icosahedral symmetry. We showed that this monodromy group is a subgroup of finite index in SL 2 (Z[ √ 5]) and the period map brings the Winger pencil to a curve on the Hilbert modular surface SL 2 (Z[ √ 5])/H 2 .

The sextic plane curves that are invariant under the standard action of the icosahedral group on the projective plane make up a pencil of genus ten curves (spanned by a sum of six lines and three times a conic). This pencil was first considered in a note by R. M. Winger in 1925 and is nowadays named after him. The second author recently gave this a modern treatment and proved among other things that it contains essentially every smooth genus ten curve with icosahedral symmetry. We here show that the Jacobian of such a curve contains the tensor product of an elliptic curve with a certain integral representation of the icosahedral group. We find that the elliptic curve comes with a distinguished point of order 3, which proves that the monodromy on this part of the homology is the full congruence subgroup Γ1(3)⊂SL2(Z)$\Gamma _1(3)\subset \text{SL}_2(\mathbb {Z})$ and subsequently identify the base of the pencil with the associated modular curve. We also observe that the Winger pencil “accounts” for the deformation of the Jacobian of Bring's curve as a principal abelian fourfold with an action of the icosahedral group.