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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.

14Q05 Computational aspects of algebraic curves
14H40 Jacobians, Prym varieties
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