On hyperelliptic abelian functions of genus 3 (Q533418)

Let \(X\) be a hyperelliptic curve of genus \(3\), \(J(X)\) the Jacobian of \(X\), \(\Theta\) the theta divisor, \(A\) the affine ring of \(J(X)\setminus\Theta\), and \(\mathcal{D}\) the ring of holomorphic differential operators on \(J(X)\). The purpose of this paper is to determine the structure of \(A\) as a \(\mathcal{D}\)-module. This is achieved by proving a conjecture on the \(\mathcal{D}\)-free resolution of \(A\) for hyperelliptic curves of arbitrary genus that was presented in the article [\textit{A. Nakayashiki} and \textit{F. A. Smirnov}, Commun. Math. Phys. 217, No. 3, 623--652 (2001; Zbl 0987.14021)]. Klein's hyperelliptic sigma function is used to define a certain filtration on \(A\). Explicit generators of the graduated ring of \(A\) are determined, and an explicit \(\mathcal{D}\)-free resolution of \(A\) is derived.