Summary:  This thesis consists of two main parts. The first part concerns zero cycles on abelian varieties and their relation to some Milnor type Kgroups. In chapter 1 we recall some basic properties of Milnor Kgroups and their generalizations, the Somekawa Kgroups. The main result of the first part is presented in chapter 2, where we construct, for an abelian variety A over a field k, a decreasing filtration {Fr}r≥0 of the group CH0(A) having the property that the successive quotients Fr/F r+1 are isomorphic after [special characters omitted] to a Somekawa type Kgroup. We then focus on the case when the base field is a finite extension of Q p. Using the above filtration, we prove some results of arithmetic interest about the structure of the albanese kernel, the kernel of the cycle map to etale cohomology and the BrauerManin pairing. The results of this chapter are gathered in one paper, [14]. Chapter 3 serves as a bridge between the first and the second part of this thesis. In this chapter we work with smooth quasiprojective varieties, introducing Suslin's singular homology group and Wiesend's tame class group. The latter group is a first generalization in higher dimensions of the generalized Jacobian varieties of a smooth projective curve. Using these two geometric invariants, we generalize the main theorem of chapter 2 for semiabelian varieties. We close the chapter by providing some motivation towards a more general reciprocity theory. The second part concerns a newly developed theory about reciprocity functors introduced by Ivorra and Rulling in [20]. This theory generalizes the theory of RosenlichtSerre about local symbols on commutative algebraic groups. In particular, we will see that every reciprocity functor M has local symbols corresponding to any smooth complete curve C over a field k. These local symbols induce a complex (C). In chapter 4 we focus on the case of a smooth complete curve C over an algebraically closed field k and we compute under two assumptions the homology of the local symbol complex in terms of Kgroups of reciprocity functors. We then close the thesis by providing important examples where the assumptions are satisfied. The results of this chapter are gathered in one paper, [15].
