On the existence of abelian surfaces with everywhere good reduction

2019/03/25 by Lassina Dembélé, Dembele, Lassina
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1903.10394

openalex publication_date 2019/03/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let D ≤ 2000 be a positive discriminant such that F = Q(√(D)) has narrow class one, and A/F an abelian surface of \rm GL₂-type with everywhere good reduction. Assuming that A is modular, we show that A is either an F-surface or is a base change from Q of an abelian surface B such that \rm End_Q(B) = Z, except for D = 353, 421, 1321, 1597 and 1997. In the latter case, we show that there are indeed abelian surfaces with everywhere good reduction over F for D = 353, 421 and 1597, which are non-isogenous to their Galois conjugates. These are the first known such examples.

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