Computing Crystalline Deformation Rings via the Taylor–Wiles–Kisin Patching Method

Consider an odd prime p and a fixed 2-dimensional absolutely irreducible representation r valued in a finite field of characteristic p. Let R^k denote the non-framed fixed-determinant crystalline deformation ring of r, whose points in p-adic fields parametrize the crystalline representations of Hodge-Tate weights (0,k-1) that reduce to r modulo p. In this talk, we will discuss computing arbitrarily close approximations of R^k and, consequently, approximations of the Hilbert series of R^k/p. We will begin by explaining the background of Galois deformation theory and the significance of these rings in the modularity lifting theorems. Following this, we will discuss the main results. Finally, we will discuss the algorithm if time permits.