Title: Geometry and arithmetic of bielliptic Picard curves

Abstract: I'll describe the very pretty geometry of the curves y^3 =
x^4 + ax^2 + b, and use it to "see" the quaternionic multiplication on
their Prym varieties, giving a very explicit family of QM abelian
surfaces (sometimes called "false elliptic curves"). I'll then
describe a few recent results on the arithmetic of these surfaces: a)
a full classification of all rational torsion points in this family
and b) a proof that the average rank in the corresponding family of
Pryms is at most 3. This is based on joint work with Laga and
Laga-Schembri-Voight.