The Arithmetic-Geometric Mean and Isogenies for Curves of Higher Genus

Computation of Gauss's arithmetic-geometric mean involves iteration of a simple step, whose algebro-geometric interpretation is the construction of an elliptic curve isogenous to a given one, specifically one whose period is double the original period. A higher genus analogue should involve the explicit construction of a curve whose Jacobian is isogenous to the Jacobian of a given curve. The doubling of the period matrix means that the kernel of the isogeny should be a Lagrangian subgroup of the group of points of order 2 in the Jacobian. In genus 2 such a construction was given classically by Humbert and was studied more recently by Bost and Mestre. In this article we give such a construction for general curves of genus 3. We also give a similar but simpler construction for hyperelliptic curves of genus 3. We show that for g > 4 no similar construction exists, and we also reinterpret the genus 2 case in our setup.