The arithmetic-geometric mean and isogenies for curves of higher genus

TL;DR

This paper extends the concept of the arithmetic-geometric mean to higher genus curves by explicitly constructing isogenies between their Jacobians, focusing on genus 3 and hyperelliptic cases, and analyzing the existence of such constructions.

Contribution

It provides the first explicit construction of genus 3 curve isogenies analogous to the AGM, including hyperelliptic cases, and demonstrates the non-existence for genus 4 and higher.

Findings

Constructed genus 3 curves with Jacobians isogenous to given curves.

Proved hyperelliptic genus 3 construction is a degeneration of the general case.

Showed no similar constructions exist for genus at least 4.

Abstract

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 the hyperelliptic construction is a degeneration of the general one, and we prove that the kernel of the induced isogeny on jacobians is a lagrangian subgroup of the points of order 2. We show that for g at least 4 no similar construction exists, and we also reinterpret the genus 2 case in our setup. Our construction of these correspondences uses the bigonal and the trigonal constructions, familiar in the theory of Prym varieties.