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On the order of the reduction of a point on an abelian variety

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Consider a point of infinite order on an abelian variety over a number field. Then its reduction at any place v of good reduction is a torsion point. For most of this paper we fix a rational prime ℓ and study how the ℓ-part of this reduction varies with v. Under suitable conditions we prove various statements on this ℓ-part for all v in a set of positive Dirichlet density: for example that its order is a fixed power of ℓ, that its order is non-trivial for the reductions of finitely many points, or that its order is larger than a certain explicit value that varies with v. By similar methods we prove that for all v in a set of positive Dirichlet density the reduction of a given abelian variety possesses no non-trivial supersingular abelian subvariety.

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  1. Department of Mathematics, ETH-Zentrum, CH-8092, Zürich, Switzerland

    Richard Pink

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  1. Richard Pink
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Correspondence to Richard Pink.

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Mathematics Subject Classification (2000):14K15 (11R45)

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Pink, R. On the order of the reduction of a point on an abelian variety. Math. Ann. 330, 275–291 (2004). https://doi.org/10.1007/s00208-004-0548-8

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  • DOI: https://doi.org/10.1007/s00208-004-0548-8

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