A tower of genus 2 curves related to the Kowalewski top

Abstract

Several curves of genus 2 are known, such that the equations of motion of the Kowalewski top are linearized on their Jacobians. One can expect from transcendental approaches via solutions of equations of motion in theta-functions, that their Jacobians are isogeneous. The paper focuses on two such curves: Kowalewski's and that of Bobenko-Reyman-Semenov-Tian-Shansky, the latter arising from the solution of the problem by the method of spectral curves. An isogeny is established between the Jacobians of these curves by purely algebraic means, using Richelot's transformation of a genus 2 curve. It is shown that this isogeny respects the Hamiltonian flows. The two curves are completed into an infinite tower of genus 2 curves with isogeneous Jacobians.