Number/Representation Theory

The local Langlands correspondence for a reductive group $G$ over $\mathbb Q_p$ posits a correspondence between certain smooth representations $\pi$ of $G(\mathbb Q_p)$ and certain $G$-valued (in an appropriate sense) representations $\varphi$ of $\mathrm{Gal}(\overline{\mathbb Q_p}/\mathbb Q_p)$. While brought into clearer perspective by recent work, the precise nature of what is meant by $\varphi$ corresponding to $\pi$ has historically been somewhat vague but, there is a natural guess: the characters of $\varphi$ and $\pi$ should match. This does not quite make sense as traces for $\varphi$ take as inputs Galois-group elements $\tau$, and traces on $\pi$ take as inputs certain functions $f$ on $G(\mathbb Q_p)$. Thus, it is natural to guess that to each $\tau$ there is an $f_\tau$ and one should require that $\mathrm{tr}(\tau\mid \varphi)=\mathrm{tr}(f_\tau\mid \pi)$. While such a naive statement cannot be true, it is a conjecture of Scholze--Shin that an appropriate refinement does hold. The question then becomes: what are the functions $f_\tau$? In this talk I will try to roughly explain the definition of these functions and their relationship to Shimura varieties, the so-called Langlands--Kottwitz--Scholze method.

This talk will be somewhat informal, with as little background as possible required, so everyone is welcome!