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

Ron Donagi, Ron Livné

Research output: Contribution to journalArticlepeer-review

15 Scopus citations

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 for g > 4 no similar construction exists, and we also reinterpret the genus 2 case in our setup.

Original languageAmerican English
Pages (from-to)323-339
Number of pages17
JournalAnnali della Scuola normale superiore di Pisa - Classe di scienze
Volume28
Issue number2
StatePublished - 1999

Bibliographical note

Publisher Copyright:
© 1999 Scuola Normale Superiore. All rights reserved.

Fingerprint

Dive into the research topics of 'The Arithmetic-Geometric Mean and Isogenies for Curves of Higher Genus'. Together they form a unique fingerprint.

Cite this