Given a sextic CM field K, we can ask whether there are any simple, principally polarized abelian varieties that have complex multiplication by the ring of integers of K. Furthermore, since such an abelian variety is isomorphic to the Jacobian of a curve of genus 3, we can ask if the curve is hyperelliptic or a plane quartic.

In this talk we describe how to, starting with the period matrix of a Jacobian that has strict CM by a sextic CM field K, determine if the Jacobian is hyperelliptic, and if so, how to compute a model for the curve. We will also touch briefly on how to compute the period matrices if time allows.

This is joint work with J. Balakrishnan, S. Ionica and K. Lauter.

### Algebra Seminar

Monday, March 20, 2017 - 3:15pm

#### Christelle Vincent

University of Vermont