Galois moduli and modular Hecke algebras

The overall objective of the project HEGAL is to make progress on two related research 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 /F ) of a p-adic number field F to the mod p representation theory of p- adic reductive groups defined over F (like GLn(F )). Serre’s conjecture in its original form (proved by Khare-Wintenberger) asks when a 2-dimensional representation of Gal(Q/Q) arises from a cuspidal modular Hecke eigenform and predicts the minimal weight and the level of such a form. There are generalizations to arbitrary algebraic number fields and to reductive algebraic groups other than GL2. 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 fixed n ∈ N, parametrizes the n-dimensional p-adic representations of Gal(F /F ). The irreducible components of the special fibre of the stack can be labelled by Serre weights (irreducible mod p representations of GLn(Fq), where Fq 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 (in the sense of Herzig and Gee-Herzig-Savitt) 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 of the project HEGAL 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 Up-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 HEGAL and may have potential applications to other research fields in number theory.