The geometry and arithmetic of bielliptic Picard curves

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