Genus 3 hyperelliptic curves with CM via Shimura reciprocity

Abstract: Up to isomorphism over $C$, every simple principally polarized abelian variety of dimension 3 is the Jacobian of a smooth projective curve of genus 3. Furthermore, this curve is either a hyperelliptic curve or a plane quartic. Given a sextic CM field $K$, we show that if there exists a hyperelliptic Jacobian with CM by $K$, then all principally polarized abelian varieties that are Galois conjugated to it are hyperelliptic. Using Shimura's reciprocity law, we give an algorithm for computing approximations of the invariants of the initial curve, as well as their Galois conjugates. This allows us ton define and compute class polynomials for genus 3 hyperelliptic curves with CM.

cryptography and securityalgebraic geometrynumber theory

Audience: researchers in the topic

( chat | paper | slides | video )

Comments: The slides used in the pre-recorded session are available here.

Chairs: Wouter Castryck and Chloe Martindale

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