Langlands reciprocity and the geometry of Shimura varieties: the mod p interface

Let p be a prime. Serre's Conjecture states that certain 2-dimensional mod p manifestations of number-theoretic symmetries arise from complex analytic functions called modular forms. The refined part of Serre's Conjecture further predicts the minimal invariants (weight and level) of such a modular form. Serre's Conjecture can be interpreted as a prototypical example of "mod p Langlands reciprocity,'' with the refinement as a form of "local-global compatibility.''

Serre's Conjecture is now a theorem of Khare and Wintenberger. The equivalence between the conjecture and its refinement was already known, and is integral to their proof. The weight part had been proved by Edixhoven, who further provided an alternative formulation that uses a geometric definition of mod p modular forms and includes modular forms of weight one.

In the same spirit as the seminal works of Wiles and Taylor-Wiles, the proof of Serre's Conjecture is inextricably intertwined with that of modularity theorems (i.e., cases of "classical Langlands reciprocity''). Furthermore, local-global compatibility is not only an essential part of the thread weaving together the proofs of classical and mod p modularity results; it is also fundamental to arithmetic applications of modularity, such as to Fermat's Last Theorem and the Birch--Swinnerton-Dyer Conjecture.

The refined Serre Conjecture and its generalizations have been inspirational in the development of the p-adic Langlands Programme, which aims to place the powerful interplay above in a much broader framework. These generalizations, however, have largely had an algebraic flavor, absent the geometric perspective inherent in Edixhoven's variant. This perspective was reintroduced in the co-I's work with Sasaki, enriching the scope of Serre's Conjecture and building its connection with the mod p geometry of Shimura varieties (a class of varieties generalizing classical modular curves).

The connection has in turn produced novel insights into this geometry, as in the PI's recent work with the co-I. The overarching vision of this proposal is to develop that interface. Its primary strands aim to i) establish a new framework for mod p Langlands reciprocity, ii) produce fundamental advances in our understanding of Shimura varieties in characteristic p, and iii) expand aspects of the interface to the p-adic context.

To place the proposed research in the current landscape, we remark that

the p-adic Langlands Programme is only beginning to take shape; the proposed research introduces novel approaches and perspectives that will help build its foundations and broaden its applicability. Meanwhile, our work on Shimura varieties will lead to new insight and results in arithmetic geometry and develop symbiotic links with several other fertile ongoing lines of research, such as the emerging theory of automorphic forms on G-Zips.

Furthermore, the proposed research will enhance the UK's standing as a leading centre for research in number theory, and especially the Langlands Programme.