Bielliptic Picard curves

I’ll describe the geometry and arithmetic of the curves $y^3 = x^4 + ax^2 + b$. The Jacobians of these curves factor as a product of an elliptic curve and an abelian surface $A$. The latter is an example of a “false elliptic curve”, i.e. an abelian surface with quaternionic multiplication. I’ll explain how to see this from the geometry of the curve, and then I’ll give some results on the Mordell–Weil groups $A(\mathbb{Q})$. This is based on joint work with Laga and Laga–Schembri–Voight.