On the reduction of points on abelian varieties and tori

2009/09/25 by Antonella Perucca, Perucca, Antonella
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.0909.4806

12 pages; comments are welcome

arxiv created 2009/09/25 · arxiv updated 2009/12/01

Abstract

Let G be the product of an abelian variety and a torus defined over a number field K. Let R₁,..., R_n be points in G(K). Let l be a rational prime and let a₁,..., a_n be non-negative integers. Consider the set of primes p of K satisfying the following condition: the l-adic valuation of the order of (R_i mod p) equals a_i for every i=1,...,n. We show that this set has a natural density and we characterize the n-tuples a₁,..., a_n for which the density is positive. More generally, we study the l-part of the reduction of the points.

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