On the order of the reduction of a point on an abelian variety
type: Item
instance of: scholarly article
title: On the order of the reduction of a point on an abelian variety
zbMATH Open document ID: 1077.11046
author: Richard Pink
published in: Mathematische Annalen
publication date: 2005-02-09T00:00:00Z
full work available at URL: http://hdl.handle.net/20.500.11850/160332
review text: If \(A\) is an abelian variety over a number field \(K\) and \(a\in A(K)\) is of infinite order, then the reduction \(a_v\) of \(a\) at a place \(v\) of good reduction is torsion. What is its order? More precisely, fix a rational prime \(\ell\): if \(v\) does not divide \(\ell\), what can be said about the \(\ell\)-part of \(a_v\)? Such questions are approached here by studying the action of \(\text{ Frob}_v\) on \[ \ell^{-\infty}({\mathbb Z}a)=\left\{x\in A(\bar K)\mid \exists n\geq 0 \;\ell^n x\in {\mathbb Z}a\right\} \] which, it is shown here, determines the \(\ell\)-part of~\(a_v\). The groups \(\ell^{-\infty}({\mathbb Z}a)\) is an extension of \({\mathbb Z}[1/\ell]\) by \(A[\ell^\infty]\) so it may be studied via the Tate module of the kernel \[ T_\ell(A)=\text{ Hom}\big({\mathbb Q}_\ell/{\mathbb Z}_\ell, A[\ell^\infty]), \] the Tate module for the extension \[ T_\ell(A,a)=\text{ Hom}\big({\mathbb Q}_\ell/{\mathbb Z}_\ell, \ell^{-\infty}({\mathbb Z}a)/{\mathbb Z}a\big), \] and their Galois module structures. These structures are described in \S\S{}1--2 of the paper, where it is also shown that there is a finite extension \(L\colon K\) such that \(A_w\) has no supersingular abelian subvariety for \(w\) in a set of places of \(L\) of density~\(1\). In \S{}3 we see the connection between that \(\ell\)-part of \(a_v\) and the Galois modules. The results of \S{}4 are density results for the \(\ell\)-part of \(a_v\): for instance, if \({\mathbb Z}a\) is Zariski dense and \(b\in A[\ell^\infty]\) then for \(v\) in a set of positive density, the \(\ell\)-part of \(a_v\) is equal to the reduction of~\(b\). The general tenor is that any plausible behaviour actually occurs, and for a set of positive density. The final section draws some conclusions not depending on~\(\ell\). In particular, if \(A_i\), \(1\leq i\leq d\), are abelian varieties over \(K\), \(a_i\in A_i(K)\), and for \(v\) in a set of density~\(1\) the reduction of at least one \(a_i\) is annihilated by a pow…
Mathematics Subject Classification ID: 11G10
Mathematics Subject Classification ID: 11R45
Mathematics Subject Classification ID: 14K15
zbMATH DE Number: 2132480
zbMATH Keywords: abelian variety
zbMATH Keywords: torsion
zbMATH Keywords: Tate module
reviewed by: G. K. Sankaran
MaRDI profile type: Publication
OpenAlex ID: W1983127114
DOI: 10.1007/S00208-004-0548-8
description: scientific article; zbMATH DE number 2132480