Galois moduli and modular Hecke algebras

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The overall objective is to make progress on two related programs and conjectures in arithmetic geometry: the mod p local Langlands correspondence and the generalized Serre modularity conjecture. The conjectured mod p local Langlands correspondence should relate the mod p representation theory of the absolute Galois group Gal_F of a p-adic number field F to the mod p representation theory of p-adic reductive groups defined over F (like GL_n(F)). A central object in both the Langlands and the Serre conjecture is a certain Galois moduli stack, recently introduced in work of Emerton-Gee. This is a formal algebraic stack which, for a fixed natural number n, parametrizes the n-dimensional p-adic representations of Gal_F. The irreducible components of the special fibre of the stack can be labelled by Serre weights (irreducible mod p representations of GL_n(F_q), where F_q denotes the residue field of F), but its local mod p geometry remains mysterious. On the one hand, it is expected to be the correct stack of L-parameters on the Galois side of the Langlands correspondence. On the other hand, it provides an extremely useful geometrization of the generalizations of the weight part of Serre's conjecture. It is generally believed that a precise understanding of the mod p geometry of the Emerton-Gee stack may lead to a breakthrough in these two central conjectures in number theory. The concrete objective is to analyze the mod p geometry of the Emerton-Gee stack. The main novelty hereby is the use of modular Hecke algebras (such as Iwahori-Hecke algebras, Hecke DGA's or modular U_p-operators) to describe local models for portions of the stack (related to p-adic Hodge theoretic data) or to produce comparison morphisms with familiar objects from geometric representation theory (Vinberg monoids, Satake parameters, Springer fibres). Such comparison morphisms allow for a functorial construction of interesting classes of sheaves (or complexes thereof) on the stack, whose invariants (support, cohomology etc.) may contain important arithmetic information. The necessary tools and methods from algebra and geometric representation theory will partly be developed within the project and may have potential applications to other research fields in number theory, e.g. to automorphic forms, Iwasawa theory and the study of Shimura varieties.