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The Picard-Fuchs equations for complete hyperelliptic integrals of even order curves, and the actions of the generalized Neumann system. (English) Zbl 1298.14037

The authors find the Picard-Fuchs type equations for the case of the family of even order genus 2 curves \[ \Gamma_h=\{ \omega^2=(z-a_1)(z-a_2)(z-a_3)(z^3+h_1z+h_2)\} \] which appear in quadratures of an integrable generalization of the Neumann system with a separable quartic potential.

MSC:

14H70 Relationships between algebraic curves and integrable systems
34M03 Linear ordinary differential equations and systems in the complex domain
34M56 Isomonodromic deformations for ordinary differential equations in the complex domain
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
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References:

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[17] This integral is slightly different from the canonical integral E(k), for this reason we use the notation \documentclass[12pt]{minimal}\( \begin{document}\bar{E}(k)\end{document} \).
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