[Paper Review] The arithmetic-geometric mean and isogenies for curves of higher genus

Ron Donagi, Ron Livné|arXiv (Cornell University)|Dec 21, 1997

Algebraic Geometry and Number Theory3 references19 citations

TL;DR

This paper constructs a genus-3 analogue of Gauss's arithmetic-geometric mean via isogenies between Jacobians of curves, using the bigonal and trigonal constructions to explicitly build a curve whose Jacobian is isogenous to that of a given curve, with the kernel of the isogeny being a Lagrangian subgroup of the 2-torsion points. It proves that such a construction fails for genus $ g \geq 4 $, generalizing the genus-2 case via Prym theory.

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 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.

Motivation & Objective

• To extend Gauss’s arithmetic-geometric mean construction to higher genus curves, specifically genus 3.
• To provide an explicit construction of a curve whose Jacobian is isogenous to that of a given curve via a Lagrangian 2-torsion kernel.
• To show that such a construction is impossible for genus $ g \geq 4 $, using moduli space and monodromy arguments.
• To reinterpret the classical genus-2 Humbert construction as an instance of the bigonal construction.
• To demonstrate that the hyperelliptic genus-3 construction arises as a degeneration of the general construction.

Proposed method

• Uses the bigonal and trigonal constructions from Prym variety theory to build correspondences between curves.
• Constructs a double cover of a singular curve with ordinary double points, ensuring the Prym variety is principally polarized.
• Identifies the kernel of the isogeny on Jacobians as a Lagrangian subgroup of the 2-torsion points via Weil pairing properties.
• Applies the period map and Torelli’s theorem to relate moduli spaces of curves and principally polarized abelian varieties.
• Uses monodromy and density arguments in $ \mathrm{Sp}(2g,\mathbb{R}) $ to show that the Torelli locus cannot be invariant under certain isogenies for $ g \geq 4 $.
• Analyzes the deformation type of the construction in characteristic not 2 or 3, generalizing results beyond characteristic 0.

Experimental results

Research questions

• RQ1 Can an analogue of Gauss’s arithmetic-geometric mean be constructed for curves of genus 3 using isogenies of Jacobians with Lagrangian 2-torsion kernels?
• RQ2 Does the genus-3 construction generalize the classical Humbert construction in genus 2 via the bigonal construction?
• RQ3 Is there a hyperelliptic special case of the genus-3 construction that degenerates from the general one?
• RQ4 Why does such a construction fail for curves of genus $ g \geq 4 $, despite the existence of Lagrangian subgroups in the 2-torsion?
• RQ5 Can the Torelli locus be preserved under isogenies induced by Lagrangian subgroups in higher genus, and what does this imply for moduli space geometry?

Key findings

• A construction of a curve of genus 3 whose Jacobian is isogenous to that of a given curve via a Lagrangian subgroup of the 2-torsion points is explicitly given using the trigonal and bigonal constructions.
• The kernel of the induced isogeny on Jacobians is proven to be a Lagrangian subgroup of the 2-torsion points, satisfying the required symplectic duality condition.
• The hyperelliptic genus-3 case is shown to be a degeneration of the general genus-3 construction, preserving the isogeny structure.
• The genus-2 Humbert construction is reinterpreted as an instance of the bigonal construction, with the kernel being a Lagrangian subgroup of the 2-torsion.
• For genus $ g \geq 4 $, no such isogeny construction exists because the Torelli locus would have to be dense in the moduli space of principally polarized abelian varieties, which contradicts dimension-theoretic constraints.
• The monodromy group generated by the isogeny and the modular group is dense in $ \mathrm{Sp}(2g,\mathbb{R}) $, implying that invariance of the Torelli locus under such isogenies would force it to be dense, which is impossible for $ g > 3 $.