On the reduction of points on abelian varieties and tori

Details On the reduction of points on abelian varieties and tori On the reduction of points on abelian varieties and tori

2009-09-25

arxiv.org/abs/0909.4806v1

Mathematics

Number Theory

12 pages; comments are welcome

Scientific paper

Let G be the product of an abelian variety and a torus defined over a number field K. Let R_1,..., R_n be points in G(K). Let l be a rational prime and let a_1,..., 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 a_1,..., 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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