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Uniformization of jacobi varieties of trigonal curves and nonlinear differential equations

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Abstract

We obtain an explicit realization of the Jacobi and Kummer varieties for trigonal curves of genusg (gcd(g,3)=1) of the form

$$y^3 = x^{g + 1} + \sum\limits_{\alpha ,\beta } {^\lambda 3\alpha + \left( {g + 1} \right)\beta ^{x^\alpha } y^\beta } ,0 \leqslant 3\alpha + \left( {g + 1} \right)\beta< 3g + 3,$$

as algebraic subvarieties in ℂ4g+δ, where δ=2(g−3[g/3]), and in ℂg(g+1)/2. We uniformize these varieties with the help of ℘-functions of several variables defined on the universal space of Jacobians of such curves. By way of application, we obtain a system of nonlinear partial differential equations integrable in trigonal #x2118;-functions. This system in particular contains the Boussinesq equation.

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Authors
  1. V. M. Buchstaber
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  2. V. Z. Enolskii
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  3. D. V. Leykin
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Additional information

Moscow State University, NASU Institute of Magnetism, Kiev, NASU Institute of Magnetism, Kiev. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 34, No. 3, pp. 1–16, July–September, 2000.

Translated by D. V. Leykin

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Buchstaber, V.M., Enolskii, V.Z. & Leykin, D.V. Uniformization of jacobi varieties of trigonal curves and nonlinear differential equations. Funct Anal Its Appl 34, 159–171 (2000). https://doi.org/10.1007/BF02482405

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  • DOI: https://doi.org/10.1007/BF02482405

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