Jacquet-Langlands correspondences for mod p and p-adic Hilbert modular forms
Supervisor: Dr Shu Sasaki
Project description:
In 1995, A. Wiles, in collaboration with R. Taylor, proved Fermat’s Last Theorem (FLT). This was one of the landmark results in the Langlands program, and Wiles’ ideas continue to inspire active research in many related areas of mathematics. The heart of Wiles’ proof of FLT is a modular lifting theorem which proves a case of the Langlands correspondence, via deforming a ‘mod p’ Langlands correspondence (in the sense of J. P. Serre). This modular lifting argument was subsequently gen- eralised by F. Calegari and D. Geraghty in 2018, who proposed a variant that is contingent upon various natural ‘conjectures’, but applicable nonetheless to new settings (e.g. coherent cohomology complex) which were hitherto considered unattainable.
Since 2009, I have been collaborating with F. Diamond to develop a (mod p) geometric theory of Hilbert modular forms, in terms of coherent cohomology. Our work was motivated by J. P. Serre’s speculation about a mod p analogue of the Langlands program, but it also sheds light on key problems that feed into Calegari-Geraghty’s proposed program of research. The project aims at making fundamental progress on these strands of research. More specifically,
Objective 1: building on my recent joint work on a ‘mod p Jacquet-Langlands relation’, prove a modular lifting theorem in the potentially Barsotti-Tate case following Calegari-Geraghty’s ap- proach.
Objective 2: use the ‘patching complex’ constructed in Objective 1 to prove a mod p analogue of the Jacquet-Langlands correspondence (proved in 1970 in the pioneering work of R. Langlands and H. Jacquet), from the optic of the Breuil-Mézard conjecture.
Objective 3: making appeal to insights we will have gained in Objective 2, prove a p-adic Jacquet-Langlands correspondence for each stratum of the Hilbert modular variety of Iwahori level, and thereby prove C. Breuil’s conjectures about p-adic Hilbert modular forms.
Further information: