Ordinary primes for abelian surfaces
type: Item
instance of: scholarly article
title: Ordinary primes for abelian surfaces
zbMATH Open document ID: 1336.11048
author: William F. Sawin
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2016-05-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/1506.04784
review text: An abelian variety \(A\) over a number field \(K\) is expected to have ordinary reduction at infinitely many places of \(K\). This is known to be true for \(A\) of dimension at most \(2\), and for such \(A\) the set of ordinary places is known to have a positive (natural) density. The goal of the paper under review is to compute this density in the case of an abelian surface. The author achieves a full classification, proving in particular that in the case \(\dim A = 2\) the density of the set of places of ordinary reduction is \(1\), \(\frac{1}{2}\), or \(\frac{1}{4}\), and that either of the latter two possibilities implies that \(A\) has an isogeny factor whose base change to the algebraic closure \(\overline{K}\) has complex multiplication.
Mathematics Subject Classification ID: 11G10
Mathematics Subject Classification ID: 11G15
zbMATH DE Number: 6579915
zbMATH Keywords: abelian surfaces
zbMATH Keywords: non-ordinary primes
zbMATH Keywords: good reduction
zbMATH Keywords: elliptic curves
MaRDI profile type: Publication
OpenAlex ID: W2964072080
arXiv ID: 1506.04784
cites work: Q4095018
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DOI: 10.1016/J.CRMA.2016.01.025
description: scientific article