The geometry and arithmetic of bielliptic Picard curves

Abstract

We study the geometry and arithmetic of the curves $C:y^3=x^4+a{x}^2+b$ and their associated Prym abelian surfaces $P$. We prove a Torelli theorem in this context and give a geometric proof of the fact that $P$ has quaternionic multiplication (QM) by the quaternion order of discriminant $6$. This allows us to describe the Galois action on the geometric endomorphism algebra of $P$. As an application, we classify the torsion subgroups of the Mordell-Weil groups $P(\mathbb{Q})$, as both abelian groups and $\text{End}(P)$-modules.

37 pages, minor changes

Cited by in corpus (1)

• Rational torsion points on abelian surfaces with quaternionic multiplication