On the occasion of the Compositio Mathematica prize we will have a symposium on **November 24th, 2023**, at the university of Amsterdam, with 5 speakers. At the end of the day, Guy Henniart and (posthumously) Colin Bushnell will receive the prize for their paper Local Langlands correspondence and ramification for Carayol representations.

Time | Speaker |
---|---|

10:00 | Maarten Solleveld |

11:15 | Wushi Goldring |

12:15 | Lunch |

13:30 | Benjamin Schraen |

14:45 | Pol van Hoften |

16:00 | Prize award |

16:15 | Guy Henniart |

** Maarten Solleveld**: * Towards a local Langlands correspondence in depth zero*.
Among the representations of a reductive \(p\)-adic group \(G\), those in depth zero are the ones most closely related to representations of reductive groups over finite fields. We will discuss a local Langlands correspondence for irreducible complex depth zero \(G\)-representations, that is considerably easier than in positive depth. We will see that for large classes of such representations, an explicit, constructible LLC is known. We will also look at some ideas on how to generalize this to all irreducible depth zero representations, which is joint work in progress with Tasho Kaletha.

** Wushi Goldring**: * Propagating algebraicity via functoriality*.
Some automorphic representations are known to have algebraic Hecke eigenvalues for geometric reasons. For most types of automorphic representations over number fields, the algebraicity of the Hecke eigenvalues is not known. In some cases the eigenvalues are conjectured to be algebraic, while in others they are conjectured to be transcendental. There seems to be no conjectural characterisation of when the Hecke eigenvalues should be algebraic.
We study which automorphic representations \(\pi\) admit a Langlands functoriality transfer \(\pi'\) along an \(L\)-morphism with finite kernel such that (1) \(\pi'\) is known to have algebraic Hecke eigenvalues for geometric reasons or (2) \(\pi'\) was previously conjectured to have algebraic Hecke eigenvalues. This propagates the algebraicity from \(\pi'\) to \(\pi\). In the negative direction, we give several obstructions to the existence of \(\pi'\), based both on the infinitesimal character and on the `complex conjugation part' of the archimedean Langlands parameter. In particular, this gives a conceptual explanation for why \(\pi'\) doesn't exist when \(\pi\) arises from a Maass form. In the positive direction, we exhibit new cases of algebraicity of Hecke eigenvalues for automorphic representations for which no direct link to geometry is known. In some of these cases, we also associate the Galois representations predicted by the Langlands correspondence.

**Benjamin Schraen**: *On the \(p\)-adic Langlands Program*. My goal is to give an introduction to the \(p\)-adic Langlands Program and, in a second time, to discuss some more recent results and issues.

** Pol van Hoften**: *On Exotic Hecke correspondences*. The goal of this talk is to explain joint work in progress with Jack Sempliner on the construction of "exotic" Hecke correspondences between the mod \(p\) fibers of different Shimura varieties of Hodge type. Our work generalizes forthcoming work of Xiao-Zhu; our results cover the new situation where the groups underlying the two different Shimura varieties are allowed to be to be non-isomorphic at \(p\). As a consequence of our main results, we obtain exotic isomorphisms of Igusa varieties in the style of Caraiani-Tamiozzo.

** Guy Henniart**: *On some mysteries in the local Langlands correspondence for GL\((n)\).* Let \(F\) be a \(p\)-adic field. The Langlands correspondence (LLC) for GL\((n,F)\) generalizes class field theory, which is the case where \(n=1\). If \(E/F\) is a finite Galois extension and \(G\) its Galois group, LLC associates to an irreducible representation \(\sigma\) of \(G\) of dimension \(n\) a representation \(\pi = \pi(\sigma)\) of GL\((n,F)\). In general, a concrete description of \(\pi\) from \(\sigma\) (or of \(\sigma\) from \(\pi\)) remains elusive. However one can study the behaviour of some invariants defined on both sides via ramification theory. That is what Bushnell and I did in the prized paper. In the talk, I hope to give a gentle introduction to the questions we answered, and the mysteries that remain.