2005/04/11 by Baranczuk, Stefan
#11G10 #19F99 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.math/0504215
In this paper we consider reduction maps rv : K2n+1(F)/CF → K2n+1(κv)l where F is a number field and CF denotes the subgroup of K2n+1(F) generated by l-parts (for all primes l) of kernels of the Dwyer-Friedlander map and maps rv : A(F)→ Av(κv)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 P1, ..., Ps, Q1, ..., Qs∈ K2n+1(F)/CF be the points of infinite order. Assume that for almost every prime l the following condition holds: for every set of positive integers m1, ..., ms and for almost every prime v m1 rv(P1)+... + ms rv(Ps)=0 implies m1 rv(Q1)+... + msrv(Qs)= 0. Then there exist αi, βi∈ ℤ ∖ \0 \ such that αi Pi+βi Qi=0 in B(F) for every i ∈ \1, ... s\. We also get an analogues result for abelian varieties over number fields. The main technical result of the paper says that if P1, ..., Ps are nontorsion elements of K2n+1(F)/CF, which are linearly independent over ℤ, then for any prime l, and for any set \k1,... ,ks\⊂ ℕ ∪ \0\, there are infinitely many primes v, such that the image of the point Pt via the map rv has order equal l^kt for every t ∈ \1, ..., s \.