2-isogenies on Jacobians of genus 3 curves

Abstract: We consider isogenies on Jacobians J of genus 3 curves with a kernel that is a maximal isotropic subgroup of the 2-torsion J[2] and confront a phenomenon that is new in genus 3: for genus 1 and 2 the codomain is generally again a Jacobian of a curve and we have an explicit construction of that curve. In the genus 3 case we only obtain that the codomain is a quadratic twist of a Jacobian.

We use a construction by Donagi-Livne, refined by Lehavi-Ritzenthaler that constructs the curve whose Jacobian is the codomain up to quadratic twist. We refine the construction further to explicitly determine this quadratic twist and use it to compute many examples. The construction requires the specification of a flag on the isogeny kernel and constructs the codomain in steps, in terms of various Prym varieties.

This is joint work with Damara Gagnier.

algebraic geometrynumber theory

Audience: researchers in the discipline

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SFU NT-AG seminar

Series comments: The Number Theory and Algebraic Geometry (NT-AG) seminar is a research seminar dedicated to topics related to number theory and algebraic geometry hosted by the NT-AG group (Nils Bruin, Imin Chen, Stephen Choi, Katrina Honigs, Nathan Ilten, Marni Mishna).

We acknowledge the support of PIMS, NSERC, and SFU.

For Fall 2025, the organizers are Katrina Honigs and Peter McDonald.

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