| Summary: | This thesis consists of two main parts. The first part concerns zero cycles on abelian varieties and their relation to some Milnor type K-groups. In chapter 1 we recall some basic properties of Milnor K-groups and their generalizations, the Somekawa K-groups. 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 K-group. 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 Brauer-Manin 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 quasi-projective 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 Rosenlicht-Serre 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 K-groups 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].
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