Constructing genus 3 hyperelliptic Jacobians with CM

Abstract

Given a sextic CM field $K$, we give an explicit method for finding all genus 3 hyperelliptic curves defined over $\mathbb{C}$ whose Jacobians are simple and have complex multiplication by the maximal order of this field, via an approximation of their Rosenhain invariants. Building on the work of Weng, we give an algorithm which works in complete generality, for any CM sextic field $K$, and computes minimal polynomials of the Rosenhain invariants for any period matrix of the Jacobian. This algorithm can be used to generate genus 3 hyperelliptic curves over a finite field ${\mathbb{F}}_p$ with a given zeta function by finding roots of the Rosenhain minimal polynomials modulo $p$.

20 pages; to appear in ANTS XII

Cited by in corpus (5)

• Modular invariants for genus 3 hyperelliptic curves
• Principally polarized squares of elliptic curves with field of moduli equal to Q
• Genus 3 hyperelliptic curves with CM via Shimura Reciprocity
• A characterization of the U(Omega,m) sets of a hyperelliptic curve as Omega and m vary
• Spanning the isogeny class of a power of an ordinary elliptic curve over a finite field. Application to the number of rational points of curves of genus $\le 4$