Hyperelliptic \(d\)-osculating covers and elliptic solutions of the KdV hierarchy. (Revêtements hyperelliptiques \(d\)-osculateurs et solutions elliptiques de la hiérarchie KdV). (French. English summary) Zbl 1118.14036
Summary: Let \(d\) be a positive integer, \(\mathbb K\) an algebraically closed field of characteristic 0 and \(X\) an elliptic curve defined over \(\mathbb K\). We study the hyperelliptic curves equipped with a projection over \(X\), such that the natural image of \(X\) in the Jacobian of the curve osculates to order d to the embedding of the curve, at a Weierstrass point. We construct \((d - 1)\)-dimensional families of such curves, of arbitrary big genus \(g\), obtaining, in particular, \((g+d - 1)\)-dimensional families of solutions of the KdV hierarchy, doubly periodic with respect to the \(d\)-th variable.
MSC:
| 14H45 | Special algebraic curves and curves of low genus |
| 14H70 | Relationships between algebraic curves and integrable systems |
| 14H40 | Jacobians, Prym varieties |
| 14H55 | Riemann surfaces; Weierstrass points; gap sequences |
| 14H30 | Coverings of curves, fundamental group |
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