國立清華大學 數學系

Abstract Let $K/k$ be a Galois extension of number fields with group $G$, and let $rho$ be a non-trivial irreducible representation of $G$ of dimension $n$. In light of Artin reciprocity, Langlands conjectured that the Artin L-function attached to $rho$ is automorphic. This is often called the Langlands reciprocity conjecture. Indeed, when $n=1$, this conjecture is exactly Artin reciprocity. Also, by the works of Langlands and many others, several significant progress has been made for $n=2$ and $n=4$. However, such a reciprocity conjecture is still open. >From their theory of automorphic induction and the properties of nilpotent groups, Arthur and Clozel established Langlands reciprocity for any nilpotent $K/k$. In this talk, we shall emphasise how to use group theory to study the Langlands reciprocity conjecture in much the same spirit of Arthur and Clozel. We shall introduce the notion of ``nearly nilpotent groups'', which can be seen as a generalisation of nilpotent groups (and abelian groups). If time permits, we will explain how to obtain Langlands reciprocity for any nearly nilpotent $K/k$.