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Covers of tori: genus two. (English) Zbl 1137.14017

The paper addresses the question of describing moduli spaces of curves admitting a covering map to an elliptic curve, especially in genus \(2\), and describes some applications to integrable systems. The authors survey known results, and outline some new strategies. The moduli space \(\mathcal{M}_{1}\) of elliptic curves, the GIT approach to binary forms of degree \(k\), and the moduli space \(\mathcal{M}_{2}\) of genus \(2\) curves are reviewed in sections 1.1, 1.2 and 1.3 respectively. In 1.4, the locus \(\mathcal{L}_{d}\subset \mathcal{M}_{2}\) of genus \(2\) curves admitting a degree \(d\) covering map to an elliptic curve is discussed. Section 1.5 describes genus \(2\) curves with extra automorphisms, and their relation to \(\mathcal{L}_{d}\). In 1.6, the various existing approaches for understanding the structure of \(\mathcal{L}_{d}\) are explained. In 1.7, a new approach, designed especially for counting the number of irreducible components of \(\mathcal{L}_{d}\) is outlined. In 1.8, \(\mathcal{L}_{2},\mathcal{L}_{3}\) and \(\mathcal{L}_{4}\) are investigated in more detail.
In section 2, applications to integrable systems are discussed. As is well known, it is possible to fabricate families of solutions to the KP hierarchy using an algebraic curve together with some extra data, so that the corresponding flows are linearized on the Jacobian of the curve. In 2.1, the relation between solutions of the KP hierarchy which are elliptic in the first variable (elliptic solitons), and tangential covers of an elliptic curve is discussed. Section 2.2 addresses the question of classifying the KP solutions with \(2\) variables running on the Jacobian of a curve of genus \(2\). In 2.3, co-elliptic solitons, which are KP solutions such that \(k\) of the variables run along a \(k\) dimensional Abelian subvariety of the Jacobian of a genus \(g\) curve is discussed. Finally, in 2.4, a method to equip Hurwitz spaces of covers of elliptic curves with Frobenius manifold structure is proposed.

MSC:

14H10 Families, moduli of curves (algebraic)
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
14H30 Coverings of curves, fundamental group
14H52 Elliptic curves
14H70 Relationships between algebraic curves and integrable systems

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