Jacobi inversion formulae for a curve in Weierstrass normal form

Abstract

We consider a pointed curve $(X,P)$ which is given by the Weierstrass normal form, $y^r+A_1(x)y^{r-1}+A_2(x)y^{r-2}+\cdots +A_{r-1}(x)y+A_r(x)$ where $x$ is an affine coordinate on ${\mathbb{P}}^1$, the point $\infty$ on $X$ is mapped to $x=\infty$, and each $A_j$ is a polynomial in $x$ of degree $\le js/r$ for a certain coprime positive integers $r$ and $s$ ($r<s$) so that its Weierstrass non-gap sequence at $\infty$ is a numerical semigroup. It is a natural generalization of Weierstrass' equation in the Weierstrass elliptic function theory. We investigate such a curve and show the Jacobi inversion formulae of the strata of its Jacobian using the result of Jorgenson (Israel J. Math (1992) 77 pp 273-284).

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