Skip to main content
School of Mathematical Sciences

A variation in the Breuil-Mezard conjecture and local-global compatibility for Hilbert modular forms

Supervisor: Dr Shu Sasaki

Project description:

One of the most fundamental subjects in modern algebraic number theory is the Langlands program. The Langlands program proposes what, and how, concrete objects in many different areas of mathematics– such as number theory, geometry, representation theory and analysis– are intrinsically connected. It essentially postulates a ‘dictionary’ that would allow us to translate a hard problem in one area into an ‘easier’ problem in another, where more tools are at hand to tackle the problem. Solving problems in the Langlands program, e.g. establishing cases of the ‘Langlands correspondence’, will therefore have transformative impacts on all relevant research areas. For example, A. Wiles proved Fermat’s Last Theorem in 1995, by tapping into this intra-disciplinary aspect of the Langlands program; and his ideas continue to inspire active research in the areas.

In his seminal work in 1987, J. P. Serre (a Field medallist in 1954) formulated a conjecture, known nowadays as Serre’s conjecture, and speculated that there should be a ‘mod p’ analogue of the Langlands program. In my recent joint work with F. Diamond, I have expanded on Serre’s vision and formulated a set of new conjectures, analogous to Serre’s, that synthesise existing conjectures in the area. One insight that has since emerged, as a result of the formulation of our conjecture, is that there should be a ‘non-regular’ analogue of the Breuil-Mézard conjecture about geometry of crystalline deformation rings.

Objective 1: formulate, and prove, a such conjecture in the setting of GL_2, following the calculations I made in the quadratic case.

Objective 2: use objective 1 to generalise work of Breuil-Herzig in the non-generic case in the setting of GL_2, and establish a local-global compatibility result for the p-ordinary part of the p-adic completed cohomology of the Hilbert modular variety.

Further information:

How to apply

Entry requirements

Fees and funding

Back to top