The arithmetic-geometric mean and isogenies for curves of higher genus
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Ron Donagi [1] ; Ron Livné [2]
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[1] University of Pennsylvania
University of Pennsylvania
City of Philadelphia, Estados Unidos
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[2] Hebrew University of Jerusalem
Hebrew University of Jerusalem
Israel
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- Localización: Annali della Scuola Normale Superiore di Pisa. Classe di scienze, ISSN 0391-173X, Vol. 28, Nº 2, 1999, págs. 323-339
- Idioma: inglés
- Enlaces
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Resumen
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. Our construction of these correspondences uses the bigonal and the trigonal constructions, familiar in the theory of Prym varieties
