Some Remarks on the Hyperelliptic Moduli of Genus 3

In 1967, Shioda [20] determined the ring of invariants of binary octavics and their syzygies using the symbolic method. We discover that the syzygies determined in [20] are incorrect. In this paper, we compute the correct equations among the invariants of the binary octavics and give necessary and sufficient conditions for two genus 3 hyperelliptic curves to be isomorphic over an algebraically closed field k, char k ≠ 2, 3, 5, 7. For the first time, an explicit equation of the hyperelliptic moduli for genus 3 is computed in terms of absolute invariants.