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Prym varieties and applications. (English) Zbl 1149.14045
The author surveys the theory of the Prym varieties of ramified double coverings of algebraic curves (another source of information on these varieties is the classical book “Theta functions on Riemann surfaces” [J. D. Fay, Lect. Notes Math. 352, Springer-Verlag, Berlin (1973; Zbl 0281.30013)] and their appearance in the theory of integrable systems. Therewith he discusses only finite-dimensional integrable systems, in particular, the Hénon-Heiles system, the Kowalewski rigid body motion and the Kirchhoff equations of motion of a solid body in an ideal fluid and puts aside soliton equations which are integrable in terms of Prym theta functions corresponding to ramified coverings (the most known of them is the Novikov-Veselov equation) and which were recently used by Krichever for solving the Riemann-Schottky type problems for such Prym varieties.

##### MSC:
 14Q05 Computational aspects of algebraic curves 14H40 Jacobians, Prym varieties 70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
##### Keywords:
algebraic curves; Prym varieties; integrable systems
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##### References:
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