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Abstract

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 $\{F^{r}\}_{r\geq 0}$ of the group $CH_{0}(A)$ having the property that the successive quotients $F^{r}/F^{r+1}$ are isomorphic after $\displaystyle\otimes\mathbb{Z}[\frac{1}{r!}]$ to a Somekawa type $K$-group. We then focus on the case when the base field is a finite extension of $\mathbb{Q}_{p}$. 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 \'{e}tale cohomology and the Brauer-Manin pairing. The results of this chapter are gathered in one paper, \citep{Gaz1}. 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 semi-abelian 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 R\"{u}lling in \citep{IR}. This theory generalizes the theory of Rosenlicht-Serre about local symbols on commutative algebraic groups. In particular, we will see that every reciprocity functor $\mathcal{M}$ has local symbols corresponding to any smooth complete curve $C$ over a field $k$. These local symbols induce a complex $(\underline{\underline{C}})$. 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, \citep{Gaz2}.

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