## 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 language | American English |
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Pages (from-to) | 323-339 |

Number of pages | 17 |

Journal | Annali della Scuola normale superiore di Pisa - Classe di scienze |

Volume | 28 |

Issue number | 2 |

State | Published - 1999 |

### Bibliographical note

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