Difference of solutions for the inversion problem of ultra-elliptic integrals

Abstract

Let $V$ be a hyperelliptic curve of genus 2 defined by $Y^2=f(X)$, where $f(X)$ is a polynomial of degree 5. The sigma function associated with $V$ is a holomorphic function on ${\mathbb{C}}^2$. For a point $P$ on $V$, we consider the problem to express the $X$-coordinate of $P$ in terms of the image of $P$ under the Abel-Jacobi map. Two meromorphic functions $f_2$ and $g_2$ on ${\mathbb{C}}^2$ which give solutions of this problem are known. Since $f_2$ and $g_2$ coincide on the zero set of the sigma function, it is expected that $f_2-g_2$ can be divided by the sigma function. In this paper, we decompose $f_2-g_2$ into a product of the sigma function and a meromorphic function explicitly.