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Automorphic forms, L-functions and the Langlands conjectures

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Despite the tremendous progress made during the 50 years since they were formulated, the Langlands conjectures remain mysterious until today. The conjectures, in their original form, relate finite dimensional representations $\rho$ of the absolute Galois group of $\mathbb Q$ to certain $L$-packets of automorphic representations $\pi$ of the adele groups $G({\mathbb A})$, for arbitrary reductive groups $G$ over $\mathbb Q$. This relationship should be, in a very precise way, controlled by a coincidence $L(\rho,s)=L(\pi,s)$ of $L$-series, and similarly of $\varepsilon$-factors, attached to $\rho$ respectively $\pi$. Furthermore, it is supposed that the correspondence respects the local-global principle. The analogous local Langlands conjecture, linking finite dimensional representations of the absolute Galois groups for completions $F$ of $\mathbb Q$ to smooth irreducible representations of $G(F)$, has been proved by Harris, Taylor and Henniart for the groups $G=Gl(n)$ in 2001. However, beyond this not much is known for arbitrary $G$. We give a survey on this, including a discussion of some recent results on the Langlands correspondence in the case $G=GSp(4)$.

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