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Abstract: |
In 1987, J.P. Serre formulated a set of conjectures about weights and levels of odd two-dimensional modular mod p representations of the absolute Galois group of Q. Serre's conjecture itself was proved by Khare and Wintenberger in 2009, but it has also inspired a good deal of new mathematics. One strand of research spurred on by this development is a generalisation of Serre's conjecture over to totally real number fields; and it was in the work of Buzzard, Diamond and Jarvis in 2010 that the very first attempt was made (while focusing exclusively on *regular* weights). In my joint work with Diamond, we improve on the BDJ conjectures and formulate new conjectures about *general* weights of (geometric) mod p Hilbert modular forms. I will explain what our conjectures say and demonstrate some evidence that we are on the right track. |
