Mod p Langlands program for p-adic groups and Hecke algebras

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The Langlands program, initiated in the 1960s, is a set of conjectures predicting a unification of number theory and of representation theory of groups. The numerous developments of this program under its classical form in the last 20 years have had remarkable consequences such as the proof of Fermat's last theorem, and the one of Serre's modularity conjecture.Around 2000, the question of a p-adic/mod p version of these conjectures was raised, motivated by natural questions of p-adic arithmetic geometry. Because of unexpected and poorly understood phenomena involving groups other than GL_2(Q_p), statements of a general p-adic/mod p local Langlands conjecture remain elusive.For GL_2(Q_p) however, a correspondence has been established by P. Colmez and V. Paskunas (and based on the work of many others) with spectacular consequences such as the proof of most cases of the Fontaine-Mazur conjecture (by M. Kisin, M. Emerton).Progress remains to be made for more general groups and exploring the mod p aspect of the local Langlands program is a promising approach.In previous work, the PI highlighted the role of a certain category of Hecke modules and proved the "numerical mod p Langlands correspondence for Hecke modules" for p-adic GL_n conjectured by M.-F. Vigneras. This was the first result with a mod p Langlands flavor involving at once all p-adic general linear groups without any restriction on the rank. The proposal aims at studying the mod p representation theory of p-adic reductive groups and associated Hecke algebras. At the heart of this proposal is the wish to shed a geometric light on the right terms of a potential mod p Langlands correspondence. There are 3 main proposed directions of research:1 • Explore the possibility of a mod p Langlands correspondence for Hecke modules for a general p-adic reductive group G over a p-adic field F. This project is motivated by a/ recent progress by E. Grosse-Klönne who constructed a functor from Hecke modules to a category of objects that should encode information about certain representations of the absolute Galois group of F b/ the classification by the PI of all supersingular Hecke modules c/the work by K. Koziol, under the supervision of the PI, establishing a Langlands correspondence for packets of Hecke modules for SL(n,F), which is compatible with the one for GL(n,F) and with Grosse-Klönne's functor. Based on this, it is now possible to explore a form of mod p principle of functoriality, in the context of a Langlands correspondence for Hecke modules to start with. 2 • Formalize the representation theory of affine Hecke algebras with parameter zero. This is motivated by previous work of the PI in collaboration with P. Schneider that explores the homological properties (cohomological dimensions, duality functor) of pro-p Iwahori Hecke algebras. Because of the model of GL(2,Q_p) and of Grosse-Klönne's work, it is expected that the mod p Langlands correspondence will be given by a functor though it is still unclear what are the relevant categories to consider. Introducing tools from noncommutative geometry in the study of Hecke modules will contribute to clarifying this crucial point.3 • Translate the work on Hecke modules in 1 • and 2 • in terms of the mod p representation theory of the p-adic group G and its link to Galois representations. The link between Hecke modules and representations of G is more subtle than in the setting of complex representations and should involve derived categories. Strategies towards such investigations are outlined in the proposal. This far reaching question could eventually be related to a mod p version of a geometric Satake isomorphism.