Some Remarks on the Hyperelliptic Moduli of Genus 3

In 1967, Shioda 20 Shioda , T. ( 1967 ). On the graded ring of invariants of binary octavics . Amer. J. Math. 89 : 1022 – 1046 .Crossref], Web of Science ®] , Google Scholar]] determined the ring of invariants of binary octavics and their syzygies using the symbolic method. We discover that the syzygies determined in 20 Shioda , T. ( 1967 ). On the graded ring of invariants of binary octavics . Amer. J. Math. 89 : 1022 – 1046 .Crossref], Web of Science ®] , Google Scholar]] 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.