The geometry and arithmetic of bielliptic Picard curves

TL;DR

This paper explores the geometry and arithmetic of bielliptic Picard curves and their Prym surfaces, proving a Torelli theorem, establishing quaternionic multiplication, and classifying Mordell-Weil torsion subgroups.

Contribution

It provides a Torelli theorem for these curves, demonstrates quaternionic multiplication on Prym surfaces, and classifies their Mordell-Weil torsion subgroups.

Findings

Prym surfaces have quaternionic multiplication by a specific quaternion order.

The Galois action on endomorphism algebra is explicitly described.

Classification of Mordell-Weil torsion subgroups for these Prym surfaces.

Abstract

We study the geometry and arithmetic of the curves $C \colon y^3 = x^4 + ax^2 + b$ 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.