English

The geometry and arithmetic of bielliptic Picard curves

Algebraic Geometry 2024-12-10 v3 Number Theory

Abstract

We study the geometry and arithmetic of the curves C ⁣:y3=x4+ax2+bC \colon y^3 = x^4 + ax^2 + b and their associated Prym abelian surfaces PP. We prove a Torelli theorem in this context and give a geometric proof of the fact that PP has quaternionic multiplication (QM) by the quaternion order of discriminant 66. This allows us to describe the Galois action on the geometric endomorphism algebra of PP. As an application, we classify the torsion subgroups of the Mordell-Weil groups P(Q)P(\mathbb{Q}), as both abelian groups and End(P)\text{End}(P)-modules.

Keywords

Cite

@article{arxiv.2308.15297,
  title  = {The geometry and arithmetic of bielliptic Picard curves},
  author = {Jef Laga and Ari Shnidman},
  journal= {arXiv preprint arXiv:2308.15297},
  year   = {2024}
}

Comments

37 pages, minor changes