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

Abstract

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 φ of G and a non-zero integer c such that φ(P ) = c Q . 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 -adic valuation of the order for some fixed rational prime ( -adic support problem).

The Multilinear Support Problem for Products of Abelian Varieties and Tori

“…We study the generalization of the support problem to several points, namely the multilinear support problem. This variant was first considered by Barańczuk in [2] and [3] and by the author in [10,Section 5]. We are able to generalize Larsen's result [7,Theorem 1], thus solving the multilinear support problem for products of abelian varieties and tori: Theorem 1.…”

“…The answer is in general negative for abelian varieties and for tori, as was shown respectively by Larsen in [7] and by the author and Demeyer in [6]. Nevertheless, for products of abelian varieties and tori we proved in [10] that there exist a non-zero integer c and an endomorphism φ such that φ(P ) = cQ (for abelian varieties this result is due to Larsen, see [7]).…”

“…Indeed, by the Poincaré Reducibility Theorem, G is K-isogenous to such a product, so we can reason as in [10,Lemma 7].…”

“…In this paper we are concerned with the support problem, which asks to what extent dependency relations (over the endomorphism ring) can be derived from conditions on the order of the reductions of the points. For a detailed history of the support problem and the known results, see [10] and [6].…”

“…where α is in End K (B n+1 ) and it is non-zero, and where α n+1,j is in Hom K (B j , B n+1 ) (see [10,Lemma 4]). Since α is an isogeny, let d be its degree.…”