Generalizations of Serre's Conjecture

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This project investigates some of the mod p aspects of the p-adic Langlands program, and their applications to more classical problems in number theory. Just as the Langlands program relates several areas of study (Galois representations, automorphic forms, and the representation theory of reductive groups over local and global fields), the p-adic Langlands program studies the relations between deformations of Galois representations, p-adic Hodge theory, the p-adic representation theory of reductive groups, and cohomology with p-adic coefficients. However, the p-adic program is only in an embryonic stage, and there is very little by way of general conjectures, let alone theorems. It is natural to consider the reduction mod p of the p-adic objects involved in the p-adic Langlands program, and when one does so one is led firstly to questions involving generalizations of the weight part of Serre's conjecture. The PI proposes to generalize this conjecture to arbitrary reductive groups, and to prove some special cases. The PI also proposes to prove the first cases of the Sato-Tate conjecture for modular forms of weight greater than 2.The Langlands program is a branch of mathematics that includes ideas from number theory (the study of equations in whole numbers), representation theory and analysis. The number theoretic part of the program consists of a vast set of interlinked conjectures relating the solutions of equations in whole numbers to other apparently unrelated mathematical objects. Recently, it has become clear that this part of the Langlands program should generalise to what is known as a "p-adic Langlands program". This new program does not, as yet, have a precise conjectural formulation, and the PI proposes to investigate the correct formulations of the conjectures that should make up the program, to prove cases of them, and apply them to more classical problems in number theory.