Computing equations of hyperelliptic curves whose Jacobian has CM
Le lundi 30 mai, Christelle Vincent, professeur assistant à l'Université du Vermont, donnera un séminaire le lundi 30 mai à 14h en salle 207.
It is known that given a CM sextic field, there exists a non-empty finite set of abelian varieties of dimension 3 that have complex multiplication by this field. Under certain conditions on the field and the CM-type, this abelian variety can be guaranteed to be principally polarizable and simple. This ensures that the abelian variety is the Jacobian of a hyperelliptic curve or a plane quartic curve.
In this talk, we begin by showing how to generate a full set of period matrices for each isomorphism class of simple, principally polarized abelian variety with CM by a sextic field K. We then show how to determine whether the abelian variety is a hyperelliptic or plane quartic curve. Finally, in the hyperelliptic case, we show how to compute a model for the curve (joint work with J. Balakrishnan, S. Ionica et K. Lauter).