Geodesic
Uniformization of Jacobi Varieties of Trigonal Curves and Nonlinear Differential Equations
Funkcionalʹnyj analiz i ego priloženiâ, Tome 34 (2000) no. 3, pp. 1-16 
V. M. Buchstaber; D. V. Leikin; V. Z. Ènol'skii. Uniformization of Jacobi Varieties of Trigonal Curves and Nonlinear Differential Equations. Funkcionalʹnyj analiz i ego priloženiâ, Tome 34 (2000) no. 3, pp. 1-16. http://geodesic.mathdoc.fr/item/FAA_2000_34_3_a0/
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Voir la notice de l'article provenant de la source Math-Net.Ru

We obtain an explicit realization of the Jacobi and Kummer varieties for trigonal curves of genus $g$ ($\gcd(g,3)=1$) of the form $$ y^3=x^{g+1}+\sum_{\alpha,\beta}\lambda_{3\alpha +(g+1)\beta}x^{\alpha}y^{\beta},\qquad 0\le3\alpha+(g+1)\beta 3g+3, $$ as algebraic subvarieties in $\mathbb{C}^{4g+\delta}$, where $\delta=2(g-3[g/3])$, and in $\mathbb{C}^{g(g+1)/2}$. We uniformize these varieties with the help of $\wp$-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 $\wp$-functions. This system in particular contains the oussinesq equation.

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