1997/12/21 by Ron Donagi, Donagi, Ron, Ron Livné +2 · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #Algebraic Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Nonlinear Waves and Solitons #Polynomial and algebraic computation #alg-geom #math.AG
paper · pdf · doi:10.48550/arxiv.alg-geom/9712027
Latex, 18 pages
arxiv created 1997/12/21 · openalex publication_date 1997/12/21 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.