Number Theory Seminar

When studying congruences of modular forms, Serre (1987) asked if a 'mod p Langlands philosophy' is lurking beneath and can better explain his weight recipe. He also inquired whether his recipe could be generalized to general reductive groups .In recent years, a new trend has emerged involving the study of the moduli of mod p Langlands parameters and their local models, leading to rapid advancements in this subject, particularly in the case of general linear groups.

In this talk, we will focus on more general reductive groups that are tamely ramified at p. I will present my recent work on mod p Langlands parameters (arXiv:2306.02093, 2309.00761) and propose a speculative generalization of Serre's weight recipe. If time permits, I will outline the strategy used to establish that all mod p Langlands parameters for (possibly ramified) p-adic unitary groups admit a de Rham lift, and I will explain its significance in the study of the moduli of Langlands parameters (for unitary groups).