Kisin’s lectures on the eigencurve via Galois representations

1: Overview of main results. 2: The eigencurve via Galois representations. 3: Classification of crystalline Galois representations. (2 lectures). 4: Construction of potentially semi-stable deformation ring (2 lectures). 5: Modularity of potentially Barsotti-Tate representations. 6: The Fontaine-Mazur conjecture for GL2 (2 lectures). Recall the statement of the Fontaine-Mazur conjecture, which says that a continuous two dimensional Qp-representation ρ of GQ = Gal(Q̄/Q), which is odd irreducible, unramified outside finitely many primes, and whose restriction to the decomposition group at p is potentially semi-stable, necessarily arises from a modular form. A good test case for the conjecture is when ρ comes from an overconvergent eigenform f of finite slope and integral weight k ≥ 2. Such representations do not come from true modular forms, but they can be closely approximated by representations which do. The main result of [5] is that, apart from a certain exceptional case, the Fontaine-Mazur conjecture is true for these representations. To prove this one shows that any finite slope, overconvergent eigenform f admits a crystalline period with Frobenius acting by the Up eigenvalue λ. This is a kind of p-adic interpolation of a result of Saito [10]. When the representation attached to f is potentially semi-stable, this implies that valp(λ) ≤ k − 1, and a result of Coleman [3] guarantees that f is classical, other than in a certain exceptional case. The condition that a two dimensional Galois representation has such a period can be used to define a rigid analytic space Xfs which is expected to coincide with the eigencurve. The remaining lectures will discuss progress towards the Fontaine-Mazur conjecture. There are two approaches, one which works for potentially Barsotti-Tate representations, and one which should work for essentially all potentially semistable representations, but which currently requires (among other things) that the extension over which the representation becomes semi-stable.is abelian. Both approaches rely on a classification of semi-stable Galois representations over ramified