Uniformization of Jacobi Varieties of Trigonal Curves and Nonlinear Differential Equations

V. M. Buchstaber ; D. V. Leikin ; V. Z. Ènol'skii

Funkcionalʹnyj analiz i ego priloženiâ, Tome 34 (2000) no. 3, pp. 1-16

Résumé

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.