×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.