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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

##### MSC:
 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
##### Keywords:
algebraic curves; Calogero-Moser dynamical system; Jacobian
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##### References:
 [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). [3] H. Airault, H. McKean, and J. Moser, Commun. Pure Appl. Math.,30, 95-148 (1977). · Zbl 0338.35024 [4] A. Krazer, Lehrbuch der Thetafunktionen, Teubner, Leipzig (1903); reprinted: Chelsea, New York (1970). · JFM 34.0492.08 [5] E. D. Belokolos, A. I. Bobenko, V. B. Matveev, and V. Z. ?nol’skii, Usp. Mat. Nauk,41, No. 2 (1986). [6] V. Z. ?nol’skii, Dokl. Akad. Nauk SSSR,272, No. 2, 104-109 (1984). [7] C. Hermite, Oeuvres de C. Hermite, tome III, Gauthier-Villars, Paris (1912), pp. 249-261.
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