Two variants of the support problem for products of abelian varieties and tori

Details Two variants of the support problem for products of abelian varieties and tori Two variants of the support problem for products of abelian varieties and tori

2007-12-17

arxiv.org/abs/0712.2815v3

Mathematics

Number Theory

13 pages; v2 results generalized; v3 incorporated referee comments, final version to appear in Journal of Number Theory

Scientific paper

10.1016/j.jnt.2009.01.005

Let G be the product of an abelian variety and a torus defined over a number field K. Let P and Q be K-rational points on G. Suppose that for all but finitely many primes p of K the order of (Q mod p) divides the order of (P mod p). Then there exist a K-endomorphism f of G and a non-zero integer c such that f(P)=cQ. Furthermore, we are able to prove the above result with weaker assumptions: instead of comparing the order of the points we only compare the radical of the order (radical support problem) or the l-adic valuation of the order for some fixed rational prime l (l-adic support problem).

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