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Automorphic Forms and Moduli Spaces of Galois Representations

Final Report Summary - AF AND MSOGR (Automorphic Forms and Moduli Spaces of Galois Representations)

The project results in outputs in three areas: on moduli spaces of mod p representations, on the p-adic Langlands program, and on the Gouvea-Mazur conjecture. These are all part of the Langlands program, a web of conjectures relating number theory to other areas of mathematics.

One part of the project concerned the construction of moduli spaces. The first paper on this subject (with Emerton) appeared during the project, and a detailed study of the 2-dimensional spaces is being carried out with Caraiani, Emerton and Savitt; this paper is over 100 pages and almost complete, and another 50 page preprint with Emerton should appear very soon. This work was one of the main focuses of a conference in Pisa in June 2016 organised by Michael Harris and Peter Schneider.

My joint work on the p-adic Langlands program with Caraiani-Emerton-Geraghty-Paskunas-Shin, and on the Breuil-M\'ezard conjecture with Kisin, during the project, has considerably advanced the state of the art; in particular, it has made it clear that there will be a p-adic Langlands correspondence with considerable links to automorphy lifting theorems. The concrete output is a proof of the weight part of Serre's conjecture for Hilbert modular forms, and a construction of a candidate for the general n-dimensional p-adic Langlands correspondence.

Finally, considerable numerical data was gained towards a better understanding of the project on the Gouvea-Mazur conjecture, and a survey article on the problem was written with Kevin Buzzard.