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