An equivalent condition for abelian varieties over finite fields to have QM

2023/11/18 by Keisuke Arai, Arai, Keisuke, Yuuki Takai +1
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Coding theory and cryptography #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2311.11051

Abstract

In this paper, we give an equivalent condition for an abelian variety over a finite field to have multiplication by a quaternion algebra over a number field. We prove the result by combining Tate's classification of the endomorphism algebras of abelian varieties over finite fields with Yu's criterion of the existence of homomorphisms between semi-simple algebras.

Related

• The arithmetic of QM-abelian surfaces through their Galois representations
• Which quaternion algebras act on a modular abelian variety?
• The field of moduli of quaternionic multiplication on abelian varieties
• Homomorphisms of abelian varieties over geometric fields of finite characteristic
• Abelian varieties with quaternion and complex multiplication