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Dynamics of the Calogero-Moser system and the reduction of hyperelliptic integrals of elliptic integrals. (English. Russian original) Zbl 0602.14031
Funct. Anal. Appl. 20, 62-64 (1986); translation from Funkts. Anal. Prilozh. 20, No. 1, 73-74 (1986).
Let \(C=(\alpha,\lambda)\) be the algebraic curve \(\lambda^ 2-3\lambda \wp (\alpha)-\wp '(\alpha)=0\), \(\wp^{'2}=4\wp^ 3-g_ 2\wp -g_ 3\) [associated to a Calogero-Moser dynamical system by I. M. Kritschever, Funkts. Anal. Prilozh. 14, No.4, 45-54 (1980; Zbl 0462.35080)]. The curve C is birationally equivalent with the curve \(\tilde C=(z,w): w^ 2=(z^ 2-3g_ 2)(z+3e_ 1)(z+3e_ 2)(z+3e_ 3).\) Let \(\pi_ M: C\to M\) and \({\hat \pi}_{\tilde M}: C\to \tilde M\) be the covering projection over the tori \(M=(\wp,\wp ')\) and \(\tilde M=(y,v)\), \(v^ 2(y^ 2-3g_ 2)(y+9g_ 3/g_ 2)\) defined respectively by the formulas \(\wp =(z^ 3/27+g_ 3)(z^ 2/3-g_ 2)^{-1}\) and \(y=(4z^ 3-9g_ 2z)/3g_ z\). Let \(\tau_ M\) be the Jacobi parameter of M, \(\phi_ M: M\to J_{\tau_ M},\phi_{\tilde C}: \tilde C\to J_{\tau_{\tilde C}}\) the Abel maps of M and \(\tilde C\) in their Jacobian \(J_{\tau_ M}\) and \(J_{\tau_{\tilde C}}.\)
The main result of the note asserts the commutativity of a diagram constructed with the maps \(\pi_ M\), \({\hat \pi}_{\tilde M}\), \(\phi_ M\) and \(\phi_{\tilde M}\).
Reviewer: M.Ţarina

14H45 Special algebraic curves and curves of low genus
14H40 Jacobians, Prym varieties
70F99 Dynamics of a system of particles, including celestial mechanics
14H52 Elliptic curves
33E05 Elliptic functions and integrals
Full Text: DOI
[1] M. Krichever, Funkts. Anal. Prilozhen.,14, No. 4, 45-54 (1980).
[2] H. Bateman and A. Erdelyi, Higher Transcendental Functions, Vol. III, McGraw-Hill, New York (1955).
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