Abelian varieties that split modulo all but finitely many primes

2024/04/12 by Enric Florit, Florit, Enric
Computer Science · Mathematics · #11G10 #11R52 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2404.08496

openalex publication_date 2024/04/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let A be a simple abelian variety over a number field k such that End(A) is noncommutative. We show that A splits modulo all but finitely many primes of k. We prove this by considering the subalgebras of End(A_{\mathfrak p})⊗ℚ which have prime Schur index. Our main tools are Tate's characterization of endomorphism algebras of abelian varieties over finite fields, and a Theorem of Chia-Fu Yu on embeddings of simple algebras.

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