$K$-theory, chamber homology and base change for the \lowercase{p}-adic groups $SL(2)$, $GL(1)$ and $GL(2)$.

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Abstract

The thrust of this thesis is to describe base change $BC_{E/F}$ at the level of chamber homology and K-theory for some \lowercase{p}-adic groups, such as $SL(2, F)$, $GL(1, F)$ and $GL(2, F)$. Here $F$ is a non-archimedean local field and $E$ is a Galois extension of $F$. We have had to master the representation theory of $SL(2)$ and $GL(2)$ including the Langlands parameters.\newline\indent The main result is an explicit computation of the effect of base change on the chamber homology groups, each of which is constructed from cycles. This will have an important connection with the Baum-Connes correspondence for such p-adic groups. This thesis involved the arithmetic of fields such as $E$ and $F$, geometry of trees, the homology groups and the Weil group $\mathcal{W}_{F}$.

Keyword(s)

Algebraic Topology.; Base Change.; Chamber Homology.; K-Theory.; Non-Commutative Geometry.; Number Theory.; Representation Theory.

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