I propose to solve three problems. The first is to prove Serre’s conjecture for real quadratic fields. I will do this by using automorphic induction to transfer the problem to U(4) over the rational numbers, where I will use automorphy lifting theorems and results on the weight part of Serre's conjecture that I established in my earlier work to reduce the problem to improving results in small weight and level. I will prove these base cases via integral p-adic Hodge theory and discriminant bounds.
The second problem is to develop a geometric theory of moduli spaces of mod p and p-adic Galois representations, and to use it to establish the Breuil–Mezard conjecture in arbitrary dimension, by reinterpreting the conjecture in geometric terms, independently of any fixed mod p Galois representation. This will transform the subject by building the first connections between the p-adic Langlands program and the geometric Langlands program, providing an entirely new world of techniques for number theorists. As a consequence of the Breuil-Mezard conjecture, I will be able to deduce far stronger automorphy lifting theorems (in arbitrary dimension) than those currently available.
The third problem is to prove a strengthened version of the Gouvea–Mazur conjecture, by completely determining the reduction mod p of certain two-dimensional crystalline representations. I will do this by means of explicit computations with the p-adic local Langlands correspondence for GL_2(Q_p), as well as by improving existing arguments which prove multiplicity one theorems via automorphy lifting theorems. This work will show that the existence of counterexamples to the Gouvea-Mazur conjecture is due to a purely local phenomenon, and that when this local obstruction vanishes, far stronger conjectures of Buzzard on the slopes of the U_p operator hold.
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SW7 2AZ London
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