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

In this paper we consider reduction maps $r_{v} : K_{2n+1}(F)/C_{F} \to K_{2n+1}(\kappa_{v}){l}$where$F$isanumberfieldand$C{F}$ 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 $P_1,\mathrm{.}\mathrm{.}\mathrm{.},P_s,Q_1,\mathrm{.}\mathrm{.}\mathrm{.},Q_s\in K_{2n+1}(F)\mathrm{/}{C}_F$ be the points of infinite order. Assume that for almost every prime $l$ the following condition holds: for every set of positive integers $m_1,\mathrm{.}\mathrm{.}\mathrm{.},m_s$ and for almost every prime $v$ $$m_1{r}_v(P_1)+\mathrm{.}\mathrm{.}\mathrm{.}+m_s{r}_v(P_s)=0\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{e}\mathrm{s}{m}_1{r}_v(Q_1)+\mathrm{.}\mathrm{.}\mathrm{.}+m_s{r}_v(Q_s)=0.$$ 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 $i\in \{1,\mathrm{.}\mathrm{.}\mathrm{.}s\}$. We also get an analogues result for abelian varieties over number fields. The main technical result of the paper says that if $P_1,\mathrm{.}\mathrm{.}\mathrm{.},P_s$ are nontorsion elements of $K_{2n+1}(F)\mathrm{/}{C}_F$, 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}$,thereareinfinitelymanyprimes$v$, such that the image of the point $P_t$ via the map $r_v$ has order equal $l^{k_t}$ for every $t\in \{1,\mathrm{.}\mathrm{.}\mathrm{.},s\}$.