zbMATH — the first resource for mathematics

Sigma, tau and abelian functions of algebraic curves. (English) Zbl 1223.14067
Given an algebraic curve $$X_g$$ of genus $$g$$ with its Riemann period matrix $$\tau$$, the fundamental abelian functions on the Jacobi variety $$\text{Jac}(X_g)$$ can be realized as the second logarithmic derivatives of theta functions $$\theta(u,\tau)$$, $$u\in \text{Jac}(X_g)$$, $\wp_{ij}(u)= -\dfrac{\partial^2\ln\theta(u,\tau)}{\partial u_i\partial u_j}.$ In this paper, the authors study the abelian functions associated with $$(n,s)$$-curves given by $X_g: \; y^n=x^s +\sum_{ni+sj< ns} \lambda_{ij}x^iy^j,$ comparing and contrasting three different methods to construct differential relations between them. The simplest case corresponds to the elliptic curves of equation $$y^2=4x^3-g_2x-g_3$$, where for the Weierstrass $$\wp$$-function and $$x=\wp(u)$$, $$y=\wp'(u)$$, the differential relations are $\wp''=6\wp^2-\frac{g_2}{2}, \quad {\wp'}^2=4\wp^3-g_2\wp -g_3.$ They conclude that the classical method, based on expressing the $$\sigma$$-function in terms of the $$\theta$$-function, is more effective than the two methods associated with the $$\tau$$-function. Nevertheless, according to the authors, these last methods, one focused on Plücker relations and the other on residues, are more systematic and may provide a suitable way for computing PDEs associated with a wider range of curves.

MSC:
 14Q05 Computational aspects of algebraic curves 14H40 Jacobians, Prym varieties
Full Text: