On abelian surfaces with potential quaternionic multiplication

2004/09/20 by Luis Dieulefait, Luís Dieulefait, Dieulefait, Luis +3
Mathematics · #11G18 #14G35 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic and Geometric Analysis #FOS: Mathematics #Mathematics and Applications #Number Theory (math.NT) #math.AG #math.NT #msc:11G18 #msc:14G35

paper · pdf · doi:10.48550/arxiv.math/0409357

To appear in Bull. Belgian Math. Soc

arxiv created 2004/09/20 · openalex publication_date 2004/09/20 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

An abelian surface A over a field K has potential quaternionic multiplication if the ring End_ K (A) of geometric endomorphisms of A is an order in an indefinite rational division quaternion algebra. In this brief note, we study the possible structures of the ring of endomorphisms of these surfaces and we provide explicit examples of Jacobians of curves of genus two which show that our result is sharp.

Related

• The arithmetic of QM-abelian surfaces through their Galois representations
• Abelian surfaces with an automorphism and quaternionic multiplication
• The field of moduli of quaternionic multiplication on abelian varieties
• On the non-commutative endomorphism rings of abelian surfaces
• Endomorphism algebras of geometrically split genus 2 Jacobians over Q