An explicit formula for the genus 3 AGM

Abstract

Given a smooth non-hyperelliptic curve C of genus 3 and a maximal isotropic subgroup (w.r.t. the Weil pairing) L in Jac(C)[2], there exists a smooth curve C' s.t. Jac(C')=Jac(C)/L. This construction is symmetric. i.e. if we start with C' and the dual flag on it, we get C. A previous less explicit approach was taken by Donagi and Livne. The advantage of our construction is that it is explicit enough to describe the isomorphism H^0(C,K_C)=H^0(C',K_C').