The arithmetic-geometric mean and isogenies for curves of higher genus
| dc.creator | Donagi, Ron | |
| dc.creator | Livne, Ron | |
| dc.date | 1997-12-21 | |
| dc.date.accessioned | 2026-07-07T01:51:26Z | |
| dc.date.available | 2026-07-07T01:51:26Z | |
| dc.description | 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. | |
| dc.description | Latex, 18 pages | |
| dc.identifier | https://arxiv.org/abs/alg-geom/9712027 | |
| dc.identifier | http://arxiv.org/abs/alg-geom/9712027 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/299 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Algebraic Geometry | |
| dc.title | The arithmetic-geometric mean and isogenies for curves of higher genus | |
| dc.type | text |