#### Vol. 4, No. 1, 2020

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Genus 3 hyperelliptic curves with CM via Shimura reciprocity

### Bogdan Adrian Dina and Sorina Ionica

Vol. 4 (2020), No. 1, 161–178
##### Abstract

Up to isomorphism, every three-dimensional simple principally polarized abelian variety over $ℂ$ is the Jacobian of a smooth projective curve of genus 3. Furthermore, this curve is either a hyperelliptic curve or a plane quartic. To define hyperelliptic class polynomials, we note that given a hyperelliptic Jacobian with CM, all principally polarized abelian varieties that are Galois conjugated to it are hyperelliptic. Using Shimura’s reciprocity law, we then compute approximations of the invariants of the initial curve, as well as their Galois conjugates. We show examples of class polynomials computed using this method for the Shioda and Rosenhain invariants.

##### Keywords
hyperelliptic curve, complex multiplication, theta constants, class field
##### Mathematical Subject Classification 2010
Primary: 11G10, 11G15, 11G30