Characterizing abelian varieties by the reduction of the Mordell–Weil group

Chris Hall and Antonella Perucca

Vol. 265 (2013), No. 2, 427–440

DOI: 10.2140/pjm.2013.265.427

Abstract

Let A be an abelian variety defined over a number field K. Let p be a prime of K of good reduction and A_p the fiber of A over the residue field k_p. We call A(K)_p the image of the Mordell–Weil group via reduction modulo p, which is a subgroup of A_p(k_p). We prove in particular that the size of A(K)_p, by varying p, encodes enough information to characterize the K-isogeny class of A, provided that the following necessary condition holds: the Mordell–Weil group A(K) is Zariski dense in A. This is an analogue to a 1983 result of Faltings, considering instead the size of A_p(k_p).

Keywords

abelian varieties, Mordell–Weil group

Mathematical Subject Classification 2010

Primary: 11G05, 11G10

Milestones

Received: 22 November 2011

Revised: 6 May 2013

Accepted: 14 May 2013

Published: 28 August 2013