On reduction maps and support problem in K-theory and abelian varieties

Abstract

In this paper we consider orders of images of nontorsion points by reduction maps for abelian varieties defined over number fields and for odd dimensional K-groups of number fields. As an application we obtain the generalization of the support problem for abelian varieties and K-groups.

On a Hasse principle for Mordell–Weil groups

“…Let us also assume that there is a free Z-module L such that R ⊂ End Z (L) and for each l there is an isomorphism L ⊗ Z l ∼ = T l such that the action of R on T l comes from its action on L. Abelian varieties are principal examples of Mordell-Weil R systems satisfying all the requirements stated above with R = End F (A). Then Theorem 1.1 generalizes also for Mordell-Weil R systems satisfying the above assumptions because we can apply again Theorem 2.9 of [2] and Theorem 5.1 of [4].…”

“…Let us consider Mordell-Weil R systems which are associated to families of l-adic representations ρ l : G F → GL(T l ) such that ρ l (G F ) contains an open subgroup of homotheties. Since Theorem 2.9 of [2] and Theorem 5.1 of [4] were proven for Mordell-Weil R systems, then Proposition 2.8 and its proof generalize for the Mordell-Weil R systems. This shows that Theorem 2.9 of [2], which is stated for Mordell-Weil R systems, holds with a = 1.…”

“…The application of these results comes by referring to [1,2,4,14] where Kummer Theory and the results of [5,7,17,18,20] are key ingredients.…”

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

“…For the multiplicative group or simple abelian varieties and assuming condition (LSP), equivalent results were proven respectively by Khare in [6,Proposition 3] and by Barańczuk in [2,Theorem 8.2].…”

On a generalization of the support problem of Erdös and its analogues for abelian varieties and K-theory

“…(Assumption (A1) is fulfilled by [1,Theorem 5.1], and Assumption (A2) by [2,Lemma 3.11]). In this paper we will prove Theorem 1.1.…”