In this paper we consider reduction maps $r_{ v} : K_{ 2n+1}(F)/C_{ F} \to K_{ 2n+1}(\kappa_{ v})_{ l}$ where $F$ is a number field and $C_{ F}$ denotes the subgroup of $K_{ 2n+1}(F)$ generated by $l$-parts (for all primes $l$) of kernels of the Dwyer-Friedlander map and maps $r_{ v} : A(F)\to A_{ v}(\kappa _{ v})_{ l}$ where $A(F)$ is an abelian variety over a number field. We prove a generalization of the support problem of Schinzel for $K$-groups of number fields: Let $P_{ 1}, ..., P_{ s}, Q_{ 1}, ..., Q_{ s}\in K_{ 2n+1}(F)/C_{ F}$ be the points of infinite order. Assume that for almost every prime $l$ the following condition holds: for every set of positive integers $m_{ 1}, ..., m_{ s}$ and for almost every prime $v$ $$m_{ 1} r_{ v}(P_{ 1})+... + m_{ s} r_{ v}(P_{ s})=0 \mathrm{ implies} m_{ 1} r_{ v}(Q_{ 1})+... + m_{ s}r_{ v}(Q_{ s})= 0. $$ Then there exist $\alpha_{ i}$, $\beta_{ i}\in \mathbb{ Z} \setminus \{ 0 \}$ such that $\alpha_{ i} P_{ i}+\beta_{ i} Q_{ i}=0$ in $B(F)$ for every $i \in \{ 1, ... s\}$. We also get an analogues result for abelian varieties over number fields. The main technical result of the paper says that if $P_{ 1}, ..., P_{ s}$ are nontorsion elements of $K_{ 2n+1}(F)/C_{ F}$, which are linearly independent over $\mathbb{ Z}$, then for any prime $l$, and for any set $\{ k_{ 1},... ,k_{ s}\}\subset \mathbb{ N} \cup \{ 0\}$, there are infinitely many primes $v$, such that the image of the point $P_{ t}$ via the map $r_{ v}$ has order equal $l^{ k_{ t}}$ for every $t \in \{ 1, ..., s \}$.
Categories: math.NT math.AG