The geometry and arithmetic of bielliptic Picard curves

We study the geometry and arithmetic of the curves (Formula presented.) and their associated Prym abelian surfaces (Formula presented.). We prove a Torelli-type theorem in this context and give a geometric proof of the fact that (Formula presented.) has quaternionic multiplication by the quaternion order of discriminant 6. This allows us to describe the Galois action on the geometric endomorphism algebra of (Formula presented.). As an application, we classify the torsion subgroups of the Mordell–Weil groups (Formula presented.), as both abelian groups and (Formula presented.) -modules.