The arithmetic of QM-abelian surfaces through their Galois representations

2003/03/13 by Luís Dieulefait, Luis V. Dieulefait, Dieulefait, Luis V. +3
Computer Science · Mathematics · #11G18 #14G35 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #math.AG #math.NT #msc:11G18 #msc:14G35

paper · pdf · doi:10.48550/arxiv.math/0303162

To appear in J. Algebra

openalex publication_date 2003/03/13 · arxiv created 2004/06/30 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This note provides an insight to the diophantine properties of abelian surfaces with quaternionic multiplication over number fields. We study the fields of definition of the endomorphisms on these abelian varieties and the images of the Galois representations on their Tate modules. We illustrate our results with several explicit examples.

Related

• Galois action on the ℚ-isogeny classes of abelian surfaces with quaternionic multiplication
• On abelian surfaces with potential quaternionic multiplication
• An equivalent condition for abelian varieties over finite fields to have QM
• K-varieties and Galois representations
• The field of moduli of quaternionic multiplication on abelian varieties